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

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 Abstract This paper focuses on the calibration and assessment of cellular automata model for urban growth modeling The basic model design is a function of multitemporal satellite imagery and population density A number of transition rules considering the most influential urbanization variables are introduced in the cellular automata Such variables include land use, road networks, and population density The cellular model transition rules are calibrated both spatially and temporally to determine the optimal rule values to ensure the modeling accuracy Spatially, the model is calibrated per township (about 3x3 square miles each) such that spatial variability of the urban growth process can be taken into account Temporal calibration is performed by using a sequence of remote sensing images from which land use information at different years is extracted For assessment purpose, the multitemporal imagery is divided into two sets: training and testing data Training data are used for model calibration and testing data for evaluating the prediction results The proposed evaluation measures include fitness (for urban level match) and two types of modeling errors (for urban pattern match), based on which the optimal rules with closest modeling results to reality are selected The study shows that the use of images reduces the need for a large number of input data and hence the modeling uncertainty either from input data or through propagation Evaluation on the rule variogram reveals that the transition rule values are correlated spatially and vary with the urbanization level The paper reports the study outcome over city Indianapolis, Indiana for the past three decades using Landsat TM images and population data Modeling results, on the calibration and prediction sides, show close match with reality for both urban level and pattern quality measures This close match is a result of spatial calibration that takes into account the specific urbanization nature of each spatial unit in the modeling process Modeling results as compared to reality show more connectivity and smoothness as well Keywords: Cellular Automata; Thematic Imagery; Calibration; Urban Modeling INTRODUCTION Excessive progress has been achieved in urban dynamic modeling to understand the urban growth process (Meaille and Wald 1990; Batty and Xie 1994a and 1994b) Some urbanization models focus more on the physical aspects of the urban growth process (Wilson, 1978), while others on social factors (Jacobs, 1961) An example of the physical models is the land use transition model of Alonso and Muth in landscape economics (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 which models the urbanization process for the San Francisco Bay area only (Landis, 1992) or as general models such as HILT (human-induced land transformations) where its growth rules are designed to be general enough to fit different city structures Yang and Lo (2003) classify urban dynamic models into three categories: Cellular automata (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 recent literature mainly because of its ability to model and visualise spatial complex phenomena (Takeyama and Couclelis, 1997) Urban cellular automata models perform 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 cellular automata 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 of initial conditions is defined Transition rules are implemented recursively until matching the reference historical data cellular automata has been used by Batty and Xie (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 rate exceeds or drops below a critical value Clarke and Gydos (1998) use SLEUTH to model the urban growth in San Francisco Bay region and Washington DC/Baltimore corridor Yang and Lo (2003) use SLEUTH model to simulate future urban growth in Atlanta, Georgia with different growth scenarios Wu (2002) develops a stochastic cellular automata model to simulate rural-to-urban land conversions in the city of Guangzhou, China Calibration of cellular automata models is essential to achieve accurate modeling outcome, however, it has been ignored until recent efforts to develop cellular automata as a reliable procedure for urban development simulation (Wu 2002) Calibration is meant to determine the optimal values for parameters in the transition rules so that the modeled urban growth closely matches real urban growth The difficulty of calibration is partially due to the complexity of urban development process (Batty et al 1999) Clarke et al (1997) use visual tests to establish parameter ranges, to provide initial parameter values and to check if urban pattern matches real data (Clarke et al, 1997) About a dozen of statistical measures are calculated for certain 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 cellular automata model, while neural networks (NN) are used by Li and Yeh (2001) The fact that most urban cellular automata models need large number of data input variables is not free of risk Many uncertainties show up in the simulation output These can result from the uncertainty in the input data, uncertainty propagation through the model, and the uncertainty of the model itself in term of what degree the model represents the reality 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 input data 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 automata model 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 in literature We believe that building the model over the imagery directly is more realistic as compared to cadastral maps The imagery is a rich source of information including land cover, urban extent and growth constrains (e.g water resources) This will reduce the need of having different sets of input data layers In addition, uncertainty of urban modeling that usually rises from having multiple input data layers (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 will fit the whole study area As a matter of fact, some regions in a study area may have different urbanization behavior than others Based on this understanding, we argue that 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 are calibrated to find the best values that fit the urban dynamics for each township Temporal calibration is based on multitemporal imagery and allows transition rule values to change over time to meet the variable urban pattern in time Finally, modeling results are assessed quantitatively and qualitatively Three measures (one for urban count and two for modeling errors) are introduced for this purpose Quantitatively, calibrated rules should be able to reproduce the same urban count as reality, and qualitatively they should be able to reproduce the same urban pattern The rules that produce urban count close to real imagery with minimum modeling errors are selected The system is implemented first on synthetic city to study the effect of growth factors on urban process and then expanded to model the historical urban growth of Indianapolis, Indiana over the last three decades PRINCIPLES OF CELLULAR AUTOMATA Cellular automata is originally introduced by Ulam and von Neumann in 1940s as a framework to study the behaviour of complex systems (von Neumann, 1966) It is commonly defined as a dynamical discrete system in space and time that operates on a uniform 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 being determined 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 ,    , ,    n  (1) where;  i (i = 1…n) represents the index of the neighborhood pixels Figure shows an example of a 2-D cellular automata grid system, where I=5 represents the total space of pixels in a grid of  5= 25 pixels As an example, the state of pixel  is urban and it is surrounded by a  square neighborhood This means that there are eight neighbors for  with  i , i = 1…8 The neighborhood of pixel  can be 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    IxI  Figure An example of 2-D cellular automata In Figure 1, a  neighborhood is selected for  The metric  represents the dimension of the square neighborhood region of x  x = ±1 in the x-direction and y  y = ±1 in the y-direction with a total of pixels The neighborhood state function h t ( ) is defined as: ht ( )  (vt ( ), vt (  1 ), , v t (   n )) t t t where (v ( ), v (  1 ), , v (   n )) are the states of pixel (3)  and its neighborhood pixels at time t The selected neighborhood kernel in Figure 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: v t 1 ( )  f (ht ( )) (4) where f (ht ( )) is the transition function that represents the cellular automata transition rules defined on  and its neighborhood states Typically, the transition function f (ht ( )) uses IF THEN rules over ht ( ) to identify the future state of  at time t+1 MODELING OF SYNTHETIC CITY This section implements the cellular automata principle to a synthetic city to study the effect of modeling parameters on the urban growth process It mimics the reality through introducing complex structures for an urban system Figure presents the image of 200  200 pixels used as an input to the cellular automata algorithm Six classes 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 the urban growth process Transportation system encourages and drives the urban development For example, commercial centres should have access to road network for customer’s visit and goods delivery Therefore, cellular automata rules related to road should encourage urban development for pixels near roads River and lake pixels should be constrained such that no urban growth is allowed on these locations to conserve water resources On the other hand, lakes are considered as one of the attractive factors for urban development especially residential and recreational types, so the corresponding rule needs to show this effect on urban development The pollution sources are included as one of the constraints for urban development due to their effect on the degradation of ecological system The designed cellular automata rules should discourage urban growth in such locations Based on the above considerations, 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: IF three or more of the neighborhood pixels are urban, THEN change the test pixel to urban IF one or more of the neighborhood pixels are road AND one or more are urban, THEN change the test pixel to urban IF one or more of the neighborhood pixels are lake AND one or more are urban, THEN change the text pixel to urban ELSE keep non-urban The above cellular automata rules first check the growth constraint to preserve certain land cover classes (e.g water) then test the possibility of urban development for nonurban pixels based on the urbanization level in the neighborhood Figure shows the simulated urban growth results after 0, 25, 50 and 60 growth steps with the  10 Simulated 1987 Predicted 1992 Figure Modeling errors analysis at 1987 (simulation) and 1992 (prediction) 26 To further examine the compactness between simulated and real images, Figure 10 shows a comparison using small windows (20x20 pixels) between the ground truth images at year 1987 and 1992 and their corresponding simulated (1987) and predicted (1992) images These test windows are distributed all over the images to cover different parts of the study area Two remarks can be made based on Figure 10 First, the urban pixels in the modeling results are more connected compared to the discontinuity and fragmentation in the real data This is the result of the cellular automata modeling approach where pixels are checked in a sequential way through a connected set of neighborhood Second, even if a very close urban count is achieved between the test windows, they may have different structures This indicates that, in addition to the urban count match, there should be other evaluation measures that check the urban pattern match This justifies that Type I and II measures or total error should be used as evaluation criteria 27 Figure 10 Local compactness of urban pattern in 1987 (left two) and 1992 (right two) 5.2 Distribution of transition rule values This section studies the spatial distribution of the calibrated rule values Figure 10 shows the final calibrated rule values (residential (a), commercial (b), and population density (c)) for all townships for the calibration years (1982, 1987, and 1992) Figure 11 is similar to Figure 10, but it shows the values in two-dimensional space for a better visualization to connect the rule values with their corresponding townships spatially The residential and commercial rules have smaller values at townships close to the city center (townships to 17) than other townships away from the city center As can be seen in Figure 10, the residential rule values for townships close to the city vary between to while reaching high values, such as and 8, for remote townships (e.g., townships to and 22 to 24 at years 1987 and 1992) The same observation is also clear for the commercial rules Such results suggest that the residential and commercial rule values are inversely related with the township urbanization level This is evident in Figure 12, which shows the relation between rule values and urban count, where the smaller the urban count associated with each rule (e.g., commercial) the larger the rule values Therefore, more restricted rules are needed for townships far from the city compared to the close ones to produce realistic urban count and pattern As for the population density rule, there is no clear pattern as in the residential and commercial rules 28 a Residential rule b Commercial rule c Population density rule Figure 10 Distribution of rule values over townships 29 1982 rules (Ri,Ci,Pi) 1987 rules (Ri,Ci,Pi) 1992 rules (Ri,Ci,Pi) Figure 11 Spatially distributed rule values change over townships 30 a Residential class b Commercial class c Population rule Figure 12 Rule value vs urban count at 1992 calibration year 31 5.3 Correlation of transition rule values The section will evaluate the possible correlation among the calibrated rule values This is examined by using their variogram as a function of distance between townships or distance (difference) between urban levels The variogram is a tool that quantifies the correlation of the dependant variable in terms of certain independent variables For a given lag distance (h), the variogram is computed as the average squared difference of values separated approximately by h (Statios, 2006):  (h)  where  z (u )  z (u  h)  N ( h) N ( h ) (8) (h) is the variogram value for a specified lag distance h; z (u ) and z (u  h) represent the variable for which the variogram is being computed at location u and u+h, respectively N(h) is the total number of pairs separated by h In our work, the variogram is used to study the correlation in the rule values (residential rule for all calibrated years as an example) Our evaluation is done with lag distance (h) defined first as the distance between townships and then as the urban level At each lag distance, all the townships that have the same lag within certain tolerance are listed, the differences between their residential rule values are identified and finally the average for the squared differences is calculated to find the variogram value Table lists the variogram values of residential rules for all lag distances among townships, while Figure 13 is the corresponding plot It is clear that there is a trend and certain degree of spatial correlation exits between the residential rule values as a function of the distance between the townships The closer the townships to each other, the smaller the difference between their rule values and the smaller the variogram values For townships far (> ~30km) from each other, the variogram shows 32 an overall trend with larger values, which suggests a smaller spatial correlation between the rule values among such townships Table Variogram values of residential rules Lag distance (h), km 10 13 14 15 19 20 21 22 23 27 28 29 30 31 (h) Lag distance (h), km 32 34 35 36 38 39 40 42 43 44 48 49 51 53 56 0.69 1.55 0.81 1.33 0.33 1.61 1.24 1.80 1.59 1.13 1.50 2.00 1.73 1.86 1.76 (h) 0.42 1.70 1.46 2.10 1.58 1.92 1.75 1.47 2.42 1.25 2.25 1.78 3.50 0.92 2.00 Figure 13 Rule variogram as a function of distance lag between townships in km The same variogram analysis is applied to the residential rule values versus the residential urban count, where the latter, after normalization, is used as the lag 33 distance h It represents the difference between the township residential count and the average township count A number of lag distances are defined to group townships with close lag values The calculated results in Table are used to plot the variogram in Figure 14 As is shown, the residential rule values become less similar with the increase in the difference between urban counts Townships with closer urban counts (lag distance) tend to have similar rule values, while significant difference in urban counts leads to more diverse rule values This supports our early conclusion indicating the existence of certain relationships between the township urbanization level and the corresponding calibrated rule value Table Variogram results for h (urban similarity) values Lag Lag Residential  (h ) (h) distance distance count (Rc) (h)* (h)* 900 0.96 6.33 5300 0.76 0.97 1550 0.93 1.33 6000 0.72 1.56 1750 0.92 2.17 6600 0.70 0.33 1950 0.91 1.47 7300 0.66 1.78 2500 0.88 3.03 8500 0.61 1.11 2800 0.87 0.83 9600 0.56 0.67 3100 0.86 3.00 11800 0.46 1.44 3900 0.82 1.75 13700 0.37 1.00 4300 0.80 3.5 14800 0.32 0.72 4900 0.77 1.89 16300 0.25 0.33 * h abs( Rc  21697) / 21697 ; 21697 is the average township pixel count Residential count (Rc) 34 Figure 14 Rule variogram as a function of residential count lag distance CONCLUSIONS This work presents a methodology for using cellular automata along with multitemporal imagery for urban growth modeling The use of multitemporal imagery can simplify the definition of transition rules and limit the search space for transition rule calibration This is beneficial in reducing the need for large input data and the possible modeling uncertainty (either directly from data or through error propagation) Besides, the fact that images record most of the physical land use features required for urban modeling supports such design philosophy The availability of medium resolution satellite images at a minimal cost also makes such methodology particularly significant Calibration of transition rules is necessary to produce accurate modeling results Spatiotemporal calibration introduced in this study proves to be efficient in achieving such objective The township based spatial calibration can take into account the spatial variability in urban development This succeeds in reducing the mismatch 35 between real and simulated urban patterns through assigning rule values that best fit the urbanization behavior for each township This is a beneficial addition to the current model designs The temporal calibration is necessary to refit the model to the urban growth variability over time, since some growth periods can experience excessive or slower growth rates compared to other periods Assessment methodology is crucial for accurate urban modeling and model calibration The proposed three quality measures, fitness (to check close urban count match) and two modeling errors (for urban pattern match check) prove to be very helpful in selecting the optimal rules The total error, besides fitness, as a selection criterion shows unbiased evaluation measure that balances each error count with its corresponding class Good fitness ensures that close urban level is achieved compared to the real one Minimizing modeling errors on the other hand identifies the rules that produce the realistic spatial urban structure Modeling results present clear correlation or relationship between rule values and both the geographic location and urbanization level It is found that areas off the city require more restrict rules to match the lower urban development rate Townships that are close spatially or have similar urban development level receive similar transition rule values As an ongoing study, we are focusing on improving the search method for the best calibrated rule values Instead of using the brute force method to search the entire solution space, genetics algorithm is expected to optimize the search strategy Integrating genetics 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Imagery; Calibration; Urban Modeling INTRODUCTION Excessive progress has been achieved in urban dynamic modeling to understand the urban growth process (Meaille and Wald 1990; Batty and Xie 1994a and