KSCE Journal of Civil Engineering (2008) 12(4):281-288DOI 10.1007/s12205-008-0281-z

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www.springer.com/12205

Water Engineering

Analysis of Urban Development of Haridwar, India, Using Entropy Approach

Ramakar Jha*, Vijay P. Singh**, and V. Vatsa***

Received January 25, 2008/Accepted March 18, 2008

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Abstract

Urban development is a complex process, which should be observed at various levels and in many aspects for full understanding.The pervasive problems generated by urban development have prompted, in the present work, to study the spatial extent ofurbanization in Haridwar, India, and patterns of periodic changes in urban development (systematic/random) in order to developfuture plans for (i) urbanization promotion areas, and (ii) urbanization control areas. Remote Sensing, using Indian Remote Sensing(IRS) satellite data, was used to map the spatial extent of urbanization for the 1989, 1998, 2000 and 2002 years. GeographicalInformation System (GIS) and Entropy approach, which makes use of mathematical notions related to thermodynamics and is thedisorder of organization or randomness of organization of a system, were used to study the pattern of urban development (systematicor random) in Haridwar, India during 1989-2002. The distributed entropy and relative mean entropy values were evaluatedconsidering two location factors: (i) urban development at peripheries of 1000 m each from the centre of the city (Har Ki Pauri), (ii)urban development at peripheries of 1000 m each from the highway along the upper Ganga canal. The results obtained indicatesignificant periodic urban development in Haridwar during 1989-2002, specifically after the 1998 year. However, urbandevelopments were found to be random in nature, as the distributed relative entropy values formed a zig-zag pattern for the locationfactor from the centre of the town. The application of entropy is found to be a better alternative to conventional technique.Keywords: entropy, urban development, land use change, nonpoint source pollution, point source pollution, pollution

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1. Introduction

The ability to develop the land in an urban area heavilyinfluences economic activity and the quality of life in cities(Turkstra, 1996). One direct implication of such an urban sprawlis the change in land use and land cover. Due to increasingpopulation and high economic growth in selected landscapes,rapid urban development and land use changes have occurred inIndia during recent years. Further, urban development has bothdirect and indirect impacts on water resources. Some impactsresult from the direct modification or destruction of streams,lakes and wetlands. Other impacts occur primarily offsite due tochanges in the quality and quantity of runoff from urbandevelopment and construction activities (Dreher and Price,1992). Urban areas generate both nonpoint and point sources ofcontaminants. Point sources that have an impact on surface waterinclude industrial and municipal waste discharges; those thataffect groundwater quality include leaky underground storagefacilities, as well as miscellaneous accidental spills of organic orinorganic contaminants. Groundwater contamination by volatileorganic compounds (VOCs) is more common in urban settingsdue to the heavy use of solvents and fuels. Nonpoint sources

include runoff and/or infiltration of water from roads, industrialareas, and golf courses. Contaminants include metals, industrialorganic chemicals, nutrients, and pesticides. Keeping this inview, evaluation of periodic changes in spatial extent and patternof urban development in different years is essential for planning(i) urbanization promotion areas, and (ii) urbanization controlareas. Remote sensing techniques have been utilized innumerous studies to map land use changes and spatial extent ofurban development during different time periods (Howarth,1986; Fung and LeDerw, 1987; Eastman and Fulk, 1993; Jensenet al., 1993, 1995; Li and Yeh, 1988). These techniques includeimage differencing (Toll, 1980), image rationing (Nelson, 1983),post-classification comparison (Howarth and Wickware, 1981),masking method (Pilon et al., 1988), nearest neighborhood andprincipal component analysis (Fung and LeDrew, 1987; Li andYeh, 1988). Most of the techniques have limited capability incapturing the characteristics of urban sprawl as these have beendeveloped in the context of image analysis or fractal theory(Webster, 1995; Batty and Longley, 1994).

In addition to remote sensing approach, it was found thatentropy approach can provide information on the pattern ofperiodic changes in urban development (systematic/random).

*Scientist-E1, Environmental Hydrology Division, National Institute of Hydrology, Roorkee-247667, Uttarakhand, India (E-mail: rjha@nih.ernet.in)**Professor, Dept. of Biological and Agricultural Engineering, Texas A&M University, College Station, Texas 77843-2117, USA (Corresponding Author,

E-mail: vsingh@tamu.edu)***Research Scholar, Gurukul Kangri University, Haridwar, India

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Entropy, which is a measure of disorder or randomness (Miller,1969) has been applied to a variety of practical problems indifferent fields, including physical, biological, and social sciences(Fast, 1970; Wyatt, 1967 Morowitz, 1970, Kullback 1959,Quilstler, 1953 and 1955, Buckley, 1968, Theil, 1967 and 1972,Chapman, 1977 Medyedkov, 1967; and Thakur, 1972 and 1979).Contemporary science has accepted it as one of the foundationstones for empirical research.

In the present study, the spatial extent of urbanization andpattern of periodic changes in urban development (systematic/random) has been done for Haridwar, India, which is animportant city along the banks of River Ganga, to develop futureplan for (i) urbanization promotion areas, and (ii) urbanizationcontrol areas. Remote sensing, GIS and entropy approaches wereintegrated to fulfill the objectives of the present work.

2. Study Area and Data

Haridwar is a holy city in the state of Uttarakhand in northernIndia. The total area of Haridwar is 1994.0 km2. The district ofHaridwar lies between 77o35' to 78o15' latitude and 30°03'longitude (see Fig. 1). The city is situated just beneath theShiwalik mountain ranges and is part of west Indogangetic plainwhich is composed of Pleistocene and subrecent alluviumbrought down by rivers from the Himalayan region. Thealluvium is made up of sand, silt, clay, Kankar and gravel. Thereis no void features except that the presence of rivers and nallas.Most of the rivers of Hardwar district are flowing from west toeast.

Survey of India toposheets with a scale 1:50,000 were used for

preparation of the drainage map and contour map of the studyregion (see Fig. 1). Further, the soil map and other informationwere collected from various state and central governmentagencies. To study the land-use changes in Haridwar city satellitedata for the year 1989, 1998, 2000, and 2002 were collected andanalyzed.

3. Methodology

3.1 Mapping Spatial Extent Using Remote Sensing Tech-nique

Remote sensing data are capable of detecting and measuring avariety of elements relating to the morphology of cities, such asthe amount, shape, density, textural form and spread of urbanareas (Webster, 1995; Mesev et al., 1995). In this study, IndianRemote Sensing (IRS) data for the 1989, 1998, 2000 and 2002years were used in ERDAS-Imagine software to (i) classifypixels that are classified as built-up area (urban area), (ii) waterbodies, (iii) agricultural area, (iv) forests, and (v) barren lands.Separating agricultural areas from urban areas is particularlyimportant because past studies have demonstrated its effec-tiveness in reducing commission errors in classified imageries(Griffiths, 1988; Masek et al., 2000). Vegetation can be indicatedby the simple normalized difference vegetation index (NDVI)using multi-spectral data. For Haridwar NDVI was computedusing the equation (Justice, 1986):

(1)

where Rnir and Rred are the reflectance in near infra-red and red

NDVI Rnir Rred–( )Rnir Rred+( )

--------------------------=

Fig. 1. Study Area, Drainage and Contour Maps of the Study Region

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frequency bands.Now, to detect the urban sprawl in land use/land cover changes

in the study area, the Principal Component Analysis (PCA)image enhancement technique was adopted. PCA is a powerfultechnique for the analysis of correlated multi dimensional data.The n-channel multi spectral data can be considered as n-dimensional data. The PCA builds up a new set of axes, whichare orthogonal to each other, i.e., non-correlated. The entire dataset can be represented in terms of these new axes. The data alongthe first principal component (PC) have a great variance ordynamic range than the data plotted against either of the originalaxes. The data along the second PC have far less variance. Thisis the characteristics of all principal components. The principalcomponent images can be analyzed as separate black and whiteimages or any three component images may be combined toform a color composite. These techniques are particularlyappropriate where little apriori information concerning a givenscene is available.

3.2 Systematic/Random Pattern Analysis Entropy Appro-ach Integrated with GIS3.2.1 Geographical Information System (GIS) Approach

The maps of spatial extent obtained from PCA in ERDAS-Imagine software were input to the ILWIS-GIS software domainfor performing spatial operations. It is accepted that the urbandevelopment is affected by some primary location factors, i.e.,the distance to urban centers and roads. In the present work, thedensity of urban development has been evaluated in GIS con-sidering two location factors, viz., (i) urban development atperipheries of 1000 m each from the centre of the city (Har KiPauri), (ii) urban development at peripheries of 1000 m eachfrom the highway along the upper Ganga canal. The thematiclayers of buffer zones were created using the buffer functions ofGIS and the width of each buffer was considered to be 1000 m.In spatial information systems, a buffer zone or simply buffer, isa polygon enclosing an area within a specified distance from apoint, line or polygon. Accordingly, there are point buffers, linebuffers and polygon buffers. The town of Haridwar had 15polygon buffer zones for distance from the center of the town(Har ki Pauri) and 8 polygon buffer zones for the distance fromthe road along the upper Ganga canal. The buffer zones createdfor both the location factors are shown in Figure 3. The spatialextent maps for the 1989, 1998, 2000 and 2002 years of the studyarea were overlaid onto the buffer zone maps in order to evaluatethe density of urban development in each buffer zone of 1000 mconsidering both the location factors as discussed above. Densityof urban development (%) is defined as the amount of urbandevelopment divided by the land area in each buffer zone (Yehand Li, 1998).

The distribution of the densities over the buffers was obtainedusing the cross function of ILWIS. The cross-tabulation statisticscomputed were used to compare class value areas between twothematic layers, including the number of pixels (or hectare) incommon and percentages.

3.2.2 Entropy ApproachIn a classic paper Shannon (1948) developed the concept of'

entropy and proposed a discipline of communication theorywhich focused on the study of information theory and entropy.This theory makes use of a basic mathematical notion related toFig. 2. PCA Modeling Algorithm

Fig. 3. Creation of Polygon Buffer Zones (i) from the Centre of the City (Har Ki Pauri), and (ii) from the Highway Along the Upper GangaCanal

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thermodynamics. The second law of thermodynamics states thatthermodynamic degradation is unalterable over time, e.g., aburnt log cannot be un-burnt and lukewarm water cannot beseparated distinctly into hot water and cold water. Numerousapplications of entropy in environmental and water resources hasbeen shown by Singh (2000).

The disorder, disorganization or randomness of organization ofa system is known as its entropy (Miller, 1969). So, entropy is ameasure of disorder and information is a measure of order of asystem. Since the system changes to a less organized state from ahighly organized state and to more probable states from lessprobable states, entropy is maximized and the magnitude ofentropy is described by a set of probabilities.

As an alternative to conventional nearest neighbour techniqueMedvedkov (1966) has related this notion to the problem ofsettlement pattern analysis. He has suggested that any settlementpattern has a uniform component and a random component.These components can be measured by the method used fordetecting signals in the presence of noise in information theory.Signal is analogous to the uniform component and noise to therandom component. Entropy has been used as a measure of noisein information theory and as a measure of disorder in spatialdistributions. Medvedkov (1966) suggests that a settlement patternis a composite of two superimposed sub-patterns, one randomand the other uniform. Each of these sub-patterns will have itsown mean density of points, and the two densities added togetherwill be equal to the mean density of the composite pattern.Likewise each sub-pattern has its own entropy value, and thesesummed will be equal to the entropy value of the compositepattern. However, the entropy value of a uniform pattern is zerotherefore the entropy value of the random component will beequal to that of the total pattern. If the pattern is not perfectlyuniform the entropy function is density dependent and there is noupper limit to the value of entropy. Entropies for differentpatterns can be compared precisely only when the cell count datais obtained using the same grid in the same position on the map.Therefore, for comparative studies in time or space, the size andform of the grid must not be altered.

Again, a major difference between entropy and traditionalindices of spatial dispersion is that its value is invariant with thevalue of zones and the number of observations (n) (Thomas,1981). In contrast, the Gini coefficient and the Lorenz curve,which have been widely used in geography to describe locationpatterns, are sensitive to the size and shape of the area unitsunder observation. The modifiable area units may exert asignificant influence on the results of spatial analysis and lead tothe loss of detailed information (Openshaw, 1991). However,entropy has no such problems.

3.3.3 Mathematical FormulationLet an urban area be divided into n zones and x represent a

geographical variable to characterize these zones. Then thedegree of spatial concentration or dispersion of the geographicalvariable in the i-th zone (xi) among n zones can be measured by

Shannon's entropy E (Theil, 1967; Thomas, 1981). Entropy canbe calculated as:

(2)

where is the probability or the proportion of thevariable occurring in the ith zone, and xi is the observed value ofthe variable in the ith zone, and n is the total number of zones.The value of entropy ranges from zero to log(n). If theprobability distribution of the variable is maximally concentratedin one zone, the lowest value of the entropy, zero, will beobtained. Conversely, an evenly dispersed distribution of thevariable among zones will give a maximum entropy value oflog(n).

The relative entropy can be used to scale the entropy value intothe range from 0 to 1. The relative entropy Er is (Thomas, 1981):

(3)

If the probability distribution is maximally concentrated in oneregion, Eq. (3) would yield the lowest E value of zero.Conversely, Eq. (3) would yield a maximum E value of 1 for anevenly dispersed probability distribution. Eq. (3) was utilized foranalysis in this study.

Entropy can be used to measure the distribution of ageographical variable and thus a measure of the difference inentropy between time (t+1) and (t) can be used to indicate thechange in the degree of dispersal of land development or urbansprawl (Thomas, 1981). This can be expressed as:

(4)

in which E is the change in entropy between time (t+1) and (t).The dispersal of urban areas from a town center will lead to anincrease in the entropy value. The change of entropy can be usedto identify whether land development follows a more dispersedor compact pattern of sprawl.

Using densities of different buffer zones computed in GIS, theprobability of urban development in each buffer zone of 1000mwas estimated. Thereafter, Eq. (3) was used to compute thedistributed entropy for different buffer zones and mean relativeentropy for the 1989, 1998, 2000, 2002 years.

4. Results and Discussion

4.1 Mapping Spatial Extent using Remote Sensing Tech-nique

The IRS 1C images for the 1989, 1998, 2000 and 2002 yearswere utilized to delineate the map the spatial extent of urbandevelopment periodically. The results obtained using PrincipalComponent Analysis of these multi-temporal images is shown inFigure 4. It can be seen from the figure that the area for urbandevelopment has expanded mostly along the highway. Due totopographical conditions, initially the development was near the

E pii 1=

n

∑ log 1pi----⎝ ⎠⎛ ⎞=

pi xi xii 1=

n∑⁄=

Erpilog 1 pi⁄( )

log n( )--------------------------

i 1=

n

∑=

E∆ E t 1+( ) E t( )–=

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centre of the town (Har ki Pauri) and then it moved far away,mostly along the highway. Further, the urban developmentpattern is more clearly noticed on the urban fringes or cityperipheral agricultural areas than in the city centre.

The percentage changes in the spatial extent of urbandevelopment were computed by counting number of pixelsindicating the urban area of Haridwar, India.

4.2 Systematic/Random Pattern Analysis Entropy ApproachIntegrated with GIS4.2.1 Geographical Information System (GIS) Approach

In the next step, the outcome of remote sensing analysis in theform of spatial information was transferred to the ILWIS GISsystem. In the analysis, the buffer function of GIS was used todefine polygon buffers zones for calculating entropy. The bufferzones selected corresponded to different locations in Haridwar

dominated by residential, industrial, commercial, and mixedzone development.

Prior to the computation of density in each buffer zone of thetwo location factors, it was found necessary to compare theurban development and corresponding population density dataobtained from various sources in Haridwar. The results indicate asignificant increase in the development of the urban area duringthe period 1989-2002 (Fig. 5(a)). It is found that the density ofurban development increased gradually since 1989. However,there was exponential increase in urban development ofHaridwar during 1998-2002 due to formation of Uttarakhand asa separate state, and enhanced industrial development and urbansettlements in the surrounding areas for both location factors. Itis interesting to see that in Fig. 5(b) there has been only a gradualincrease in population of Haridwar and no exponential increaseis noticed. Also, it is noticed that some new areas of urban

Fig. 4. Principal Component Analysis of Satellite Data of Haridwar

Fig. 5. Urban Development at Haridwar, India, for Two Location Factors

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settlements have emerged after the 2000 year beyond a distanceof 12 km from the center of Haridwar.

4.2.2 Entropy ApproachAs was discussed earlier, entropy can be used to indicate the

degree of urban sprawl by examining whether land developmentin a city is dispersed or compact. The larger value indicates theoccurrence of urban development. The buffer zones created inGIS for both the location factors and density of urban area ineach buffer zone were used for computation of probability andentropy values for each buffer zone.

The mean relative entropy in terms of urban development ofHaridwar for the 1989, 1999, 2000 and 2002 years were

calculated using Eq. (3) and are given in Table 1.From Table 1, it is found that in the 1998 year entropy was

Table 1. Mean Relative Entropy Values for Both Location Factors

YearLocation factor

(a) From the centre of the city (Har Ki Pauri)

(b) From the highway along Upper Ganga Canal

1989 0.64 0.55

1998 0.97 0.63

2000 0.96 0.76

2002 0.94 0.67

Fig. 6. Entropy Values in Different Buffer Zones of Haridwar Town

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highest for location factor (a) and thereafter the entropy valuestarted to slightly decrease, which is practically insignificant. Itindicates random development in urban areas. Further, if locationfactor (b) is considered, the results substantiate the resultsobtained for location factor. This also indicates that the overallurban development in Haridwar is random.

It was found necessary to evaluate distributed relative entropyvalues in each buffer zone using Eq. (3) for both location factors(Fig. 6). From the town center Har ki Pauri, entropy was found tofollow the distance decay pattern (decresing) for the 1989 year,whereas in the 1998, 2000 and 2002 years, it follows a zig-zagpattern. This indicates that random development took place inthe Haridwar town and it is essential to develop future plans for(i) urbanization promotion areas, and (ii) urbanization controlareas. Further, in the case of distance from the highway along theupper Ganga canal, entropy followed the distance decay patternfor all data sets. The results shown in Fig. 7 indicate that entropyvalues are decreasing as we move to farther distances and peopleprefer to have their industries and residences near the highway.

Comparison of entropy values for 2000 and 2002 yearsindicates that due to the formation of Uttaranchal as a separatestate, the land value has increased enormously and has causedscattered development in the 2000 year. Similarly, during the1998 year, temporary shelters were constructed on the occasionof holy festival Kumbh in Haridwar and the scattered urbansprawl in terms of entropy can be easily visualized in some of theregions beyond a distance of 5 km from the center of the town(Fig. 7). Urban development in 1998 and 2000 indicates that themajority of developments are towards dispersed developmentrather than concentrated development.

5. Conclusions

In general, the urbanization has taken place either in a radialdirection around a well-established city or linearly alonghighways. Entropy is found to be effective for analysis of urbandevelopment patterns and for future plans for (i) urbanizationpromotion areas, and (ii) urbanization control areas. It enables to

Fig. 7. Spatial and Temporal Variation of Entropy Values

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identify which town area has better spatial efficiency in landdevelopment. The criteria used for selecting location factorshaving buffer zones provide the primary basis for urbandevelopment in a town.

The results of analysis reveal that Haridwar has experiencedrandom urban development at the dawn of the new millenniumwith the lack of proper development control and management.The dispersed pattern of land development identified bycalculating entropy from multi-temporal satellites images showsa high degree of unplanned urban development and need dueattention.

References

Batty, M. and Longley, P.A. (1994). Fractal cities: A geometry of formand function, Academic Press, London.

Buckley, W. (ed.) (1968). Modern systems research for the behaviouralscientist, Aldine Publishing Company, Chicago, IL.

Chapman, G.P. (1977). Human and environmental systems: A geo-grapher’s appraisal, Academic Press, London.

Dreher, D. W. and Price, T. H. (1992). Best management practiceguidebook for urban development, Northeastern Illinois PlanningCommission, Chicago, IL.

Eastman, J.R. and Fulk, M. (1993). “Long sequence time series evalua-tion using standardized prinicple components.” PhotogrammetricEngineering and Remote Sensing, Vol. 59, No. 6, pp. 991-996.

Fast, J.D. (1970). Entropy, Philips, London.Fung, T. and LeDrew, E. (1987). “Application of principal components

analysis change detection.” Photogrammetric Engineering andRemote Sensing, Vol. 53, No. 12, pp. 1649-1658.

Griffiths, G.H. (1988). “Monitoring urban change from Landsat TM andSPOT satellite imagery by imae differencing.” Proceedings ofIGARSS 1988, Edinburgh, Scotland, pp. 493-497.

Howarth, P.J. (1986). “Landsat digital enhancement for change detec-tion in urban environment.” Remote Sensing of Environment, Vol.13, pp. 149-160.

Howarth, P.J. and Wickware, G.M. (1981). “Procedures for changedetection using Landsat digital data.” International Journal ofRemote Sensing, Vol. 2, No. 3, pp. 277-291.

Jensen, J.R., Cowen, D.J., Althausen, J.D., Narumalani, SX., andWeatherbee, O. (1993). “An evaluation of the coast watch changedetection protocol in south carolina.” Photogrammetric Engineeringand Remote Sensing, Vol. 59, No. 6, pp. 1039-1046.

Jensen, J.R., Rutchey, K., Koch, M.S., and Narumalani, S. (1995).“Inland wetland change detection in Everglades water conservationarea 2A using a time series of normalized remotely sensed data.”Photogrammetric Engineering and Remote Sensing, Vol. 61, No. 2,199-209.

Justice, C. O. (1986). “Monitoring the grassland of semi-arid Africausing NOAA- AVHRR data.” International Journal of RemoteSensing, Vol. 7, No. 11, pp. 1383-1622.

Kullback, S. (1959). Statistics and Information Theory, John Wiley andSons, New York, NY.

Li, X. and Yeh, G.A.O. (1988). “Principal component analysis ofstacked multi-temporal images for monitoring of rapid urbanexpansion in the Pearl river delta.” International Journal of RemoteSensing, Vol. 19, No. 8, pp. 1501-1518.

Masek, J.G., Lindsey, F.E., and Goward, S.N. (2000). “Dynamics ofurban growth in the Washington DC metropolitan area, 1973-1996from Landsat observations.” International Journal of RemoteSensing, Vol. 21, No. 18, pp. 3473-3486.

Medvedkov, Y.V. (1966). “Concept of entropy in settlement patternanalysis.” Papers and Proceedings of the Regional ScienceAssociation, Vol. 8, pp. 165-168.

Medvedkov, Y.V. (1967). “The regular component in Settlement Patternas shown on a map.” Soviet Geography, Vol. 8.

Mesev, T. V., Longley P. A., Batty, M., and Sie, Y. (1995). “Morphologyfrom imagery detecting and measuring the density of urban landuse.” Environment and Planning. A-27, pp. 759-780.

Miller, J.G. (1969). Living systems: Basic concepts, Ib W. Gray et al.(eds.) General Systems Theory and Psychiatry, Boston, MA.

Morwitz, H.J. (1970). Entropy for Biologists, Academic Press, NewYork, NY.

Nelson, R.F. (1983). “Detecting forest canopy change due to insectactivity using Landsat MSS.” Photogrammetric Engineering andRemote Sensing, Vol. 49, No. 9, pp. 1303-1314.

Openshaw, S. (1991). A spatial analysis research agenda, handlinggeographical information: Methodology and potential applications(I. Masser, and M. Blackmore, editors), John Wiley & Sons, Inc.,New York, NY, pp. 18-37.

Pilon, P.G., Howarth, P.J., and Bullock, R.A. (1988). “An enhancedclassification approach to change detection in semi-arid environ-ments.” Photogrammetric Engineering and Remote Sensing, Vol.54, No. 12, pp. 1709-1716.

Quastler, H. (1953). Information Theory in Biology, University ofIllinois Press, Urbana, IL.

Quastler, H. (1955). Information theory in psychology, The Free Press,Glencoe, IL.

Shannon, C.E. (1948). “A mathematical theory of communication.” BellSystem Technical Journal, Vol. 27, pp. 378-423 and 623-656.

Singh, V.P. (2000).” The entropy theory as a tool for modelling anddecision making in environmental and water resources.” Water SA,Vol. 26, No. 1, pp. 1-11.

Thakur, B. (1972). Entropy studies in Geography. Unpublished paper,Department of Geography, University of Waterloo, Canada, pp 1-79.

Thakur, B. (1979). Urban settlements in eastern india, entropy changesand pattern analysis, Concept Publishing Company, Delhi, India.

Theil, H. (1967). Economics and information theory, Amsterdam,North-Holland.

Thiell, H. (1972). Statistical decomposition analysis, Amsterdam, NorthHolland.

Thomas, R.W. (1981). Information statistics in geography, W.H.Hutchins & Sons, Norwich.

Toll, D.L. (1980). “Urban area update procedures using Landsat data.”Proceedings, American Society of Photogrammetry, RS-E-1-17.

Turkstra J. (1996). “Urban growth and land use options for lower-income groups: A case study of Villavicencio, Columbia.” ITCJournal, 1996-1, pp. 57-63.

Wyatt, P.A.H. (1967). “Energy and entropy in chemistry.” New York: St.Martins, NY, pp. 62-79.

Webster, C.J. (1995). “Urban morphological fingerprints.” Environmentsand Planning B, Vol. 22, pp. 79-297.

Yeh, A.G.O. and Li, X. (1998). “Sustainable land development modelfor rapid growth areas using GIS.” International Journal ofGeographical Information Science, Vol. 12, No. 2, pp. 169-189.