Hello,

First of all, it is necessary to clarify how entropy is understood when it comes to urban sprawl. If the sprawl  has n different types of characteristics, then one of the entropy interpretations of n * lg(n) will not depend on spatial characteristics. The second interpretation is when there are n cells, each of which can be either occupied or not. This interpretation of entropy depends on the space. 

Hello,

Shannon's entropy can be used to measure the degree of spatial concentration and dispersion. In order to calculate the Entropy value, you will have a need for statistics of built up areas in two time periods for each zones or alternatives. In the next step it is necessary to calculate Pi value that equals to the amount of built up areas in each zones divided by the total amount of built up areas. then (Ln (Pi)) value must be calculated for each row. The next step is calculating Pi *Ln (pi). The final step is calculating the Summation of row. You should repeat mentioned steps for next time period. the difference on entropy between two periods of time, shows the change in the degree of urban sprawl. 

Hello,

Everything is a little more complicated. If Pi - is the probability of building in each zone, this means that we are working with the binary variable Yes / No (built-up / not built-up). But the building can be of different types: residential, industrial, business. 

In general, during the sprawl process, residential land is expanded, so it is apart from the other built-up areas.

It depends how you define urban sprawl and what is your objective. Entropy is the measure of irregularity. If you set a basic element (house) in space, it can be located in a regular pattern (like orthogonal streets and avenues) or randomly w.r.t. other houses. If the last case is what you name urban sprawl, then you will have curly streets with some narrow bottlenecks - and as a result longer commuting distances within this urban sprawl. You can measure inefficiency of such sprawl comparing to planned solution, for example by average distance (or time) that resident spends on commuting with the city center. But I am not sure if Shannon entropy would be the best characteristic in this case.

For the data you probably should use GIS maps with street configuration.

great answer yuri

Professor Mohamad Molaei Qelichi

i am interested to know what would be the mechanism to quantify sprawl per year involving longer periods because i am investigating the link between urban sprawl and infrastructure deficiencies and then on economic inequalities. Please elaborate further which measure of sprawl will best suit this study.

I have only a vague idea of Shannon Entropy. But did you consider the use of night satellite images and image analysis (either within GIS or e.g. ImageJ)? Even by visual inspection of those images you can well monitor urban sprawl. With the software you can do much more. If you use the radiance-calibrated world image it is also possible to differentiate light emission in inner cities (https://ngdc.noaa.gov/eog/dmsp/download_radcal.html). It's easy to locate Karachi and to compare different images during time. There is also a parameter for inter-calibration to avoid error in image comparison due to different satellites and technologies. For ImageJ you find a number of edge analysis tools to remove light over-glow and global and local thresholding plugins using Shannon Entropy. Don't know whether this helps ...

Dear Akhtar

I read your paper entitled Urban sprawl, infrastructure deficiency and economic inequalities in karachi. You can analysis the urban housing areas (Pi value) in two periods in the six districts of Karachi. It will be good measure in your study. You can also use the rate of accessibility to urban infrastructure in your study area. 

Dear Mohamad Molaei Qelichi

i hope its all fine. how should i calculate built up area from 1980 to 2015. i need urban sprawl on yearly basis to link with urban infrastructure and Inequalities. what do u infer from rate of accessibility. 

Dear Akhtar

Sorry for my late reply. It will be easy to calculate the built up area from 1980 to 2015 If you have the data and statistics in format of SHP files or Excel data sets. You only need to prepare the value of these districts in relation to built up area. furthermore, the rate of accessibility means the sanctification amounts according to the to using urban infrastructure among citizens of Karachi. It will demonstrate the Inequalities. If you have any question, let me know in a private message.

You may want to use a better method than Shannon's entropy. Even though entropy has been one of the most often used metrics for the measurement of urban sprawl, its suitability in terms of requirements for measuring urban sprawl has yet been examined only recently: Nazarnia et al. (2019) examined the behaviour and suitability of entropy as a measure of urban sprawl by applying it to seven simple model landscapes and six real-world case studies. They also investigated the influence of the choice of the city center and associated translocation of zones and assess entropy with regard to 13 suitability criteria for measures of urban sprawl. Their results show that entropy is, in many cases, not sensitive to important differences between spatial patterns of built-up areas that represent different levels of urban sprawl, e.g., dispersed vs. compact spatial arrangement of built-up areas. In addition, the value of entropy is strongly affected by changes in the choice of zones within a landscape. Finally, entropy does not meet several important suitability criteria for measuring urban sprawl; it only meets 5 out of 13 suitability criteria. The authors concluded that entropy is not suitable as a measure of urban sprawl and that more suitable metrics of urban sprawl are available that should be used instead.

From the abstract of the paper "How suitable is entropy as a measure of urban sprawl?" by Nazarnia et al. in Landscape and Urban Planning 184 (2019): 32-43.

https://www.researchgate.net/publication/230738894_Peri-Urban_to_Urban_Landscape_Patterns_Elucidation_through_Spatial_Metrics

https://www.researchgate.net/publication/229826816_Empirical_patterns_of_the_influence_of_Spatial_Resolution_of_Remote_Sensing_Data_on_Landscape_Metrics

How to calculate Shannon entropy for urban sprawl ?

You may want to use a better method than Shannon's entropy for measuring urban sprawl. Even though entropy has been used for the measurement of urban sprawl, its suitability in terms of requirements for measuring urban sprawl has recently been examined: Nazarnia et al. (2019) studied the behaviour and suitability of entropy as a measure of urban sprawl by applying it to seven simple model landscapes and six real-world case studies. They also investigated the influence of the choice of the city center and associated translocation of zones and assess entropy with regard to 13 suitability criteria for measures of urban sprawl. Their results show that entropy is, in many cases, not sensitive to important differences between spatial patterns of built-up areas that represent different levels of urban sprawl, e.g., dispersed vs. compact spatial arrangement of built-up areas. In addition, the value of entropy is strongly affected by changes in the choice of zones within a landscape. Entropy does not meet several important suitability criteria for measuring urban sprawl, either; it only meets 5 out of 13 suitability criteria. The paper concluded that entropy is not suitable as a measure of urban sprawl. More suitable metrics of urban sprawl are available that should be used instead. Have a look at the full paper "How suitable is entropy as a measure of urban sprawl?" by Nazarnia et al. in Landscape and Urban Planning 184 (2019): 32-43. Best regards,

Jochen Jaeger