In the recent years, a lot of thrust in this field has been to understand and analyse the urban sprawl pattern. Various analysts have made considerable progress in quantifying the urban sprawl pattern (Theobald, 2001; Lata et al., 2001; Torrens and Alberti, 2000; Batty et al., 1999; Barnes et al., 2001). However, all these studies have come up with different methodologies in quantifying sprawl. The common approach is to consider the behaviour of built-up area and population density over the spatial and temporal changes taking place and in most cases the pattern of such sprawls is identified by visual interpretation methods. 

Defining this dynamic phenomenon with relative precision and accuracy for predicting the future sprawl is indeed a great challenge to all working in this arena. One of the basic and major challenge is quantification of such sprawl. Although different sprawl types were identified and defined there has been an inadequacy with respect to developing mathematical relationships to define them. Further as if aggravating this problem, much of the work related to studying dynamics of urban sprawl are not carried out in the developing countries, except a few. Thus, giving very little relevance to correlate the available findings in the context of developing countries. However, the negative impacts of such urban sprawls in developing countries are more severe and intense compared to that of developed countries. Typically, the developing countries are faced with an unprecedented population growth and potentially threaten vast natural resources. In such a scenario, it is definitely an exacting effort to study, characterise and model the urban sprawl phenomenon in the context of developing countries. This study is an attempt in understanding the urban sprawl phenomenon, pattern recognition and modeling studies as well. 

Urban sprawl dynamics was analysed considering some of the causal factors. The rational behind this is to identify such factors that play a significant role in the process of urbanisation. The causal factors that were considered responsible for sprawl were:

Population has been for long accepted as a key factor of urban sprawl. The percentage built-up is the proportion of the built-up area to the total area of the village. The a population density (POPADEN) is the proportion of the population in every village to the built-up area of that village. The b population density (POPBDEN) is the proportion of population in every village to the total area of that village. The b population density is often referred as population density. Since the built-up area plays an important role in the current study for the purpose of analyses, the percentage built-up, a and b population densities are computed and analysed village-wise and categorised as a sub-zone. The annual population growth rate (AGR) is computed from the population data available from 1961 for all the villages. This growth rate is used in predicting the population for 1999 and subsequent future populations. The distance from the city centres, viz. Udupi and Mangalore to each village was calculated. Thus, the effects of proximity of the cities on the urban sprawl of these sub-zones were analysed. With these causal factors identified the modeling studies were undertaken.  

In order to explore the probable relationship of percentage built-up (dependent variable) with causal factors of sprawl (population, a and b population densities, etc.), regression analyses considering linear, quadratic (order=2), and logarithmic (power law) were tried and the results are tabulated in Annexure.   

The regression analyses revealed that the population shows linear relationship (y = a*x + b) and plays a significant role in the sprawl phenomenon. The quadratic regression analyses for second order were undertaken. All the causal factors were considered and the regression was carried out for the square of causative factors (e.g. When 'y' is the dependent variable and 'x' is the independent variable, then a polynomial regression of second order will be of the form, y = a*x² + b*x + c). The quadratic regression analyses revealed that the population b density and distance from urban centre (Mangalore) have a significant role in the sprawl phenomenon. The logarithmic (power law) regression analyses were also undertaken. The same causal factors were considered and the regression was carried out for the natural logarithmic of the causative factors (e.g. When 'y' is the dependent variable and 'x' is the independent variable, then a logarithmic regression will be of the form, log y = log a + b * log x; or y = a * x^b). The logarithmic regression analyses revealed that the population b density has significant role in the sprawl phenomenon.  

The probable relationships are

PCBUILT = 0.000611*POP99 + 10.87149 (r = 0.5789)                                                  ............2

PCBUILT = 0.005651*(POPBDEN)² - 1.2*10-7*(POPBDEN) + 6.8950 (r = 0.6823) .............3

PCBUILT = -1.7953*(MANGDIST)² + 0.02593*(MANGDIST) + 36.8607 (r = 0.60).............4

 PCBUILT = -0.9027*(UDUPIDIST)² + 0.002242* (UDUPIDIST) + 15.9731 (r = 0.583).............5

 PCBUILT = 0.270 * (POPADEN)^1.6938  (r = 0.4779) .............6

 Multivariable linear regression analysis was done to assess the cumulative effect of causal factors. The multivariate regression analyses reveal that all causal factors have a significant role in the sprawl phenomenon. Probable relationships are,

PCBUILT = - 27.08 + 0.002496*POPBDEN + 0.5743*MANGDIST + 0.8139*UDUPIDIST (r = 0.719) ............7

PCBUILT = - 18.9358 - 0.00027*POP99 + 0.005452*POPBDEN + 0.3797*MANGDIST + 0.6249*UDUPIDIST  (r = 0.761)  ............8

PCBUILT =  -38.6985 - 0.00031*POP99 + 0.006024*POPBDEN - 1.677991*AGR + 0.7577*MANGDIST + 1.0346*UDUPIDIST  (r = 0.784)    ............9

PCBUILT = - 21.7633 - 0.00031*POP99 - 0.12529*AGR - 0.0004*POPADEN + 0.006417*POPBDEN + 0.5289*MANGDIST + 0.7451*UDUPIDIST (r = 0.86)  ...........10

Likely increase in the built-up area is predicted using Equation 9. To project built-up for 2020 and 2050 corresponding population was computed considering annual growth rate based on the historical population data of 1961-2001.