Modelling spatial–temporal expansion of Lilongwe City using Shannon’s entropy model through semi-dynamic environmental mapping and analysis

This study involves analysis of the urban spatial–temporal expansion of Lilongwe City from 1973 to 2020 using Shannon’s entropy model through time-series satellite mapping. Landsat images from 1973 to 2020 in a nearly 10-year interval are used to determine spatial–temporal land use/cover changes. The city is zoned into 1 km concentric rings and four pie sections to determine both directional and spatial urban expansion trends while Shannon’s entropy model is employed to determine the degree of dispersity of the city’s sprawl. A linear regression model incorporating population as an explanatory variable is then applied to predict the spatial expansion trends for Lilongwe City. Results show that the built-up area in Lilongwe City expanded by 465.4% (9.9% per year) from 1973 to 2020, making it the second-largest land use/cover type in the city, after vegetation. Consequently, vegetation cover decreased by − 32.7% (− 0.7% per year) during the same period. High relative entropy indices (> 0.9) obtained from Shannon’s entropy model indicate a dispersed urban development for the city during the entire period of study. The North-West quadrant of the city has the highest proportion of urban expansion while the North-East quadrant has the lowest proportion, in a relative sense. Regression model predictions show that the city will most likely continue to expand by the year 2023 and then increase exponentially by the year 2033, due to high population growth. The results of this study will assist city authorities to control the expansion of the city and anticipate patterns for future urban sprawl.


Introduction
Cities in many developing countries are experiencing rapid economic and population growth, leading to an accelerated spatial-temporal urban expansion, requiring constant monitoring to facilitate sustainable development and planning.The spatial-temporal expansion of a city consists of two dimensions: spatial urban expansion, which is the change of land cover from non-urban land to urban land (Batty et al. 1999) and temporal urban expansion, which describes the urban growth evolution, where urban growth at one time-period influences future urban growth (Linard et al. 2013).Due to the impacts of spatial-temporal expansion of cities on the surrounding natural environment, many techniques (e.g., Shannon's entropy model, fractal analysis, landscape metrics, and principal component analysis) have been employed to monitor and model the growth of cities.However, Shannon's entropy model is mostly preferred due to its robustness and straightforward application (Verzosa and Gonzalez 2010; Sarvestani et al. 2011).
Shannon's Entropy model is a numerical modelling approach that estimates urban expansion and analyses the discrepancies between urban growth patterns (Dhali et al. 2019).This modelling approach has been used to model spatial expansion of the city of Riyadh in Saudi Arabia (Rahman 2016), Asmara in Eritrea (Tewolde and Cabral 2011), Accra in Ghana (Osei et al. 2015), the North 24 Parganas districts in India (Dhali et al. 2019), Thika municipality in Kenya (Muiruri and Odera 2018), and the commercial districts in Turkey (Ozturk 2017).Fractal analysis determines the complexity of the shape of the city using fractal geometry, which evaluates the extent a city spatially fills its two-dimensional area and avoids the rigidity of Euclidean geometry (Ozturk 2017).Landscape metrics quantify the spatial characteristics of the urban landscape (Magidi and Ahmed 2019) while principal component analysis is a dimension reduction tool that evaluates the consistency of spatial variables, such as different land cover classes over the years (Dhali et al. 2019).
The current study is focused on Lilongwe City (Malawi).There is limited-to-no literature on the analysis and modelling of the urban expansion of Lilongwe City.Although factors that encourage the sprawl of the city and related impacts have been documented (e.g., UN 2011; Munthali and Murayama 2011;JICA 2010), there is no reported investigation and analysis of the spatial and temporal expansion of Lilongwe City.This study quantifies urban spatial expansion of Lilongwe City over a period of 47 years (1973-2020) using Shannon's Entropy model and semi-dynamic spatial mapping at a near 10-year interval.The study also determines spatial trends in Lilongwe's urban expansion to predict future growth.

Study area
Lilongwe is the capital and the largest city in Malawi, with an area of 474 km 2 and is found in the central region of the country (Fig. 1).It is the administrative and economic centre, hosting many government head offices and enterprises.Lilongwe also happens to be one of the fastest-growing cities in Africa, with a population growth rate of 3.8% between 2008 and 2018 (National Statistics Office 2019), hence the need for rapid monitoring of its spatial expansion and related urban sprawl to determine past growth trends and predict future growth pattern.
Lilongwe City was established as a trading centre for Malawi in 1906 (JICA 2010), then called Nyasaland while Zomba City in the southern region was the capital of Malawi.However, in 1975, Lilongwe became the capital and administrative city as government head offices relocated from Zomba to Lilongwe.The population grew steadily because of this urbanization and reached 98,718 in 197798,718 in (UN 2011)).To manage the growth of the city, an urban development plan was compiled, the 2030 Master Plan.This divided the city into four sectors; Old Town (commercial zone), Capitol Hill (business and administrative district), Kanengo (industrial district) and Lumbadzi (commercial zone), where development would be concentrated (JICA 2010).
Rapid urbanization occurred in Lilongwe from 2005, due to the financial resources and investments that were concentrated in the city (UN 2011).Coupled with the social and economic development of the city (JICA 2010), there was a massive influx of rural migrants.This created challenges for city authorities as Lilongwe lacked adequate housing to accommodate the influx of new residents (JICA 2010).In response to inadequate formal housing, unplanned and uncontrolled development occurred in the surrounding and vulnerable rural-urban fringes, as people resorted to informal settlements (JICA 2010).

Data source
Six GeoTIFF data products, one for each year, were downloaded from the Earth Explorer window of the United States Geological Survey (USGS).These contained raster TIFF images, of single bands and one multispectral image, and text files of ground control point coordinates.The selected Landsat images of Lilongwe City were collected in 1973 (21 September), 1984(27 August), 1993(1 June), 2003(1 March), 2013(27 August), and 2020 (30 August)  ETM + C1 Level 1 satellite sensor while the images for 2013 and 2020 were captured by Landsat 8 OLI/TIRS C1 Level-1 satellite sensor.All images satisfied the criteria that the land and scene cloud cover be less than 10% and that image be captured during the day.
The images were chosen according to the dates at which they were captured, preferably in the winter season (May-October), to obtain cloud-free images.However, the image for 2003 does not lie within the winter season as Landsat 7 ETM + images captured after the 31st of May 2003 had data gaps and stripes, hence an image before the 31st of May was selected.The city's boundary data was obtained from the Malawi Spatial Data Platform (MAS-DAP).Population statistics for Lilongwe City were obtained from the Population Division of the United Nations, where estimates of historical and current population statistics are compiled together with projections of future statistics (UN 2019).This allowed estimates of the population during the study epochs to be extracted and used in modelling the spatial-temporal expansion of Lilongwe City.

Image classification and change detection
The selected images were enhanced by adjusting the brightness, contrast, transparency, and gamma reset in the image analysis window.Different ArcGIS functions were applied to each image based on which best highlighted the difference between the urban areas and the agricultural/vegetated areas.These images were further used to create NDVI layers to highlight the urban areas.To facilitate raster calculations, the spatial resolution had to be uniform.The Landsat MSS 1-5 C1 Level-1 image for 1973 had a spatial resolution of 60 × 60 m while all other images had a resolution of 30 × 30 m.The images from 1984 to 2020 were then resampled using the Resampling tool in the Data Management toolbox and the Nearest Neighbour technique to match the resolution of the 1973 image.
A land cover scheme was derived from the Lilongwe Land Cover 2010 Scheme (MASDAP 2014) to determine the types of land use/cover in the city.The land cover scheme included: built-up area, forest, wetland, water, cropland, bare ground/shrub.This encompasses the major land uses, such as residential, commercial, and industrial zones (builtup area), agriculture (cropland), and natural habitats or open space (wetland, water, bare ground, and shrub).The enhanced multispectral images from each year were classified using the Maximum Likelihood technique, which calculates the probability of a pixel belonging to a specific class.Thereafter, the classified images were reclassified into three classes (built-up area, vegetation, and wetland) as the land use/cover types needed for the study are the built environment and surrounding natural vegetation.
The accuracy of the classification was assessed by generating random points in the raster datasets using the 'Create Accuracy Assessment Points' tool in the Spatial Analyst toolbox.The Equalized Stratified Random sampling strategy was used to generate 40 points for each land cover class.The tool created a point vector layer with an empty 'Classified' field and a populated 'Ground Truth' field, which represented the true land cover in the image, in the attribute table.The 'Update Accuracy Assessment Points' tool was then used to update the point layer by populating the 'Classified' field.The Confusion Matrix was then calculated by using the 'Classified' and 'Ground Truth' fields of the point vector layer.
Changes in the land cover classes were determined by calculating the area of each land cover class in each year.A field for 'Area' was added to the attribute tables and the value was obtained by multiplying the count of pixels that fall in each class by the area of a single pixel (3600 m 2 ).The area from previous years was then subtracted to obtain the change in the area of the three land cover classes over time, and the percentage change was calculated for each interval from 1973 to 2020.

Determination of spatial urban expansion and sprawl
To determine the spatial pattern of urban sprawl, the city was divided into concentric circles and pie sections to identify the direction in which sprawl is prominent, following similar studies in Ghana and Turkey (Osei et al. 2015;Ozturk 2017).The concentric zone analysis (at 1 km) from the city centre was combined with pie sections (North-East, South-East, South-West, and North-West) as shown in Fig. 2.
To measure the dispersion and concentration of urban sprawl, Shannon's entropy model was employed using ArcGIS.Before the model could be applied, the area of built-up land in each sectioned ring was obtained.The vector layer in Fig. 2 was used as the feature zone data in the 'Tabulate Area' tool in the Spatial Analyst toolbox, while the six raster layers were set as the feature class data.This produced a table that displayed the area of the built environment in the four quadrants of each ring.This table was joined to the attribute table of the vector layer in Fig. 2 to obtain the corresponding area of each sectioned ring.The quantity of urban growth in each block was determined by the pixel size and the number of pixels in the block (Dhali et al. 2019).The city was classified as compact or dispersed if the Shannon entropy index was close to 0 or 1, respectively.Relative Shannon entropy values for each year were obtained using Eq. 1 (Thomas 1981;Li and Yeh 2004), where H n is the relative Shannon's entropy index, n is the total number of blocks and p i is the probability of built-up area occurring within the i th block expressed in Eq. 2 (Li and Yeh 2004), where x i is the amount of built-up area in the i th block divided by the total area of the i th block.
To quantify the change in the dispersion of the built-up area, the difference in relative Shannon's entropy values between two time periods was calculated.Equation 3 was used to determine the increase or decrease in the concentration of the built-up area (Ozturk 2017), (1) where ΔH n is the difference in relative Shannon's entropy values, H n (t) is the relative Shannon's entropy value at time (t) and H n (t − 1) is the relative Shannon's entropy value at time (t − 1) .This revealed the rate and magnitude of the dispersion of urban sprawl.

Prediction of future urban spatial expansion
Linear regression was used to model the relationship between the dependent urban environment and the independent variables that promote urban sprawl, such as population size (Montgomery et al. 2012).The Analysis Tool-Pak in Microsoft Excel was used to perform the regression analysis.The Built-up area was set as the dependent Y variable, while the population estimates were set as the independent X variable.The observed built-up area from 1973 to 2020 was used in the analysis to obtain the regression statistics, coefficients, and intercept, to determine the relationship between the built-up area and population size, and the general trend of urban sprawl.Preliminary analysis of data indicates that the relationship between the set of variables is linear and takes the form presented in Eq. 4 (Montgomery et al. 2012), where y is the dependent variable, 0 is the y intercept, 1 is the regression coefficient, x is the explanatory variable, and is the random error term.The regression coefficient explains the correlation between the two variables under study.
The linear regression equation was used to predict known values of the dependent variable based on the known independent variables used in the analysis.The predicted values, ŷ , were compared to the observed values and the residuals, e , as illustrated by Eq. 5 (Montgomery et al. 2012).
To test the performance of the regression model and whether it fits the observed dependent variable, the coefficient of determination ( R 2 ) was calculated.The coefficient of determination quantifies the proportion of variation of the dependent variable that is explained by the independent variable.Its values range from 0 to 1, where values close to 0 represent a bad fit and values close to 1 represent a good fit.The p value statistic was calculated to determine if the regression coefficient was significant to the model.Values greater than 0.05, with a 95% confidence level, indicate that the corresponding coefficient is not significant to the model and may be removed, while values less than 0.05 indicate that the coefficient is statistically significant to the model.From the regression equation, the built-up area in 2023 and 2033 was predicted, by inputting the corresponding population projections.The years 2023 and 2033 were chosen (4)

Spatial-temporal distribution of land use/cover
Figure 3 shows the spatial distribution of land use/cover for 1973, 1984, 1993, 2003, 2013, and 2020.The accuracy of the classification varied across the years.The overall accuracy and Kappa Index for 1973 was 80. 8% and 0.71, respectively, 1984 (70.8% and 0.56), 1993 (88.3% and 0.83), 2003 (83.3% and 0.75), 2013 (86.7% and 0.80), and 2020 (78.3% and 0.675).A relatively high classification accuracy was observed in 1993, while a relatively lower classification accuracy was observed in 1984, which could be due to the lower number of spectral bands of the Landsat 4-5 TM sensor.It is worth noting the relatively low classification accuracy for the 2020 image, despite the raster image being obtained from the Landsat 8 OLI/TIRS sensor, which has nine spectral bands.This may be partly due to image resampling from 1984 to 2020 from a spatial resolution of 30 × 30 m to a lower resolution of 60 × 60 m to enable the effective use of the Landsat 1-5 MSS sensor (1973) in the As shown in Figs. 3 and 4, there were some urban developments in 1973, as the city was established as a trading centre, with developments being concentrated in the Capital Hill sector in the north-east quadrant, with 811.08 ha of built-up area, and the Old Town sector, which is in the south-west quadrant, with 793.44 ha.These areas were the first to be developed due to the presence of a few government offices in Capital Hill and business enterprises in Old Town.The built-up area also extended to the northernmost parts of the city, the Lumbadzi sector.The land use/cover map for 1984 reveals that built-up land began to densify from 1973 and extended into the Capitol Hill sector, immediately northeast of the centre of the city, the Lumbadzi sector in both the north-west and north-east quadrants, and the southern parts of the city.The south-west quadrant had the largest built-up area with 1302.48 ha and the south-east with 1203.12 ha, which indicates that in 1984, the built environment was sprawling rapidly in the south.The Kanengo sector, in the north-east quadrant, also began to expand, resulting in the urban land area increasing to 871.92 ha.
There was a drastic growth in the built-up area in 1993 as it sprawled into the peripheries of the north-west quadrant, increasing the area to 1830.60 ha, and further in the southwest quadrant to an area of 2320.92 ha.The vegetation cover decreased but the wetland area increased as water bodies expanded in all quadrants.There was exponential growth in the built-up area in 2003, particularly in the south-east quadrant, the built-up are increased from 1724.40 ha in 1993 to 2124.72 ha in 2003, due to the increase in rural-urban migration which led to high demand for residential housing.Urban development also expanded into the periphery of the city in the north-west (2932.92ha) and south-west (2672.28ha) directions.However, the built environment in the north-east quadrant decreased from 1487.16 ha in 1993 to 1362.96 ha in 2003, due to the expansion of wetlands during the same period by 1696.32 ha, which might have damaged building structures.
The land use/cover map for 2013 shows the densification of impervious surfaces in the urban areas, but it is also evident that there was an increase in vegetation cover in most parts of the city except for the south-east quadrant.It also reveals that there were more urban clusters along the boundary of the city, especially in the south-west (2700.72 ha) and north-west (3292.56ha).Finally, the land use/cover map for 2020 shows the recent extent of the city and that the built-up area has increased greatly since 1973.There has been infill development within the centre of the city and urban land has sprawled into the periphery of the city in all quadrants, with the most urban development occurring in the northwest (4239.00ha) and south-east quadrants (4236.48ha).However, through the years, urban growth has been less concentrated in the north-east section of the city, having an area of 3283.92 ha in 2020, because this quadrant consists of low-density suburbs.
Figure 5 shows the percentage of land use/cover type from 1973 to 2020, while Fig. 6 shows the area differences in the three land cover classes with 1973 as the reference epoch.Figure 7 shows the area differences in the three land cover classes between successive years.The percentage of the built-up area in the city increased from 5.8% in 1973 to 32.7% in 2020, making the built-up area the secondlargest land cover type in the city as the area increased by 13,137.12ha from 1973 to 2020 (Fig. 5).The smallest change in the built-up area was observed between 2003 and 2013.The change in the built-up area with reference to 1973 was 6270.12 ha in 2003 and 7518.96ha in 2013, meaning an increase of 1248.84 ha in 10 years (Fig. 6).The vegetation cover decreased from 85.5% in 1973 to 57.5% in 2020 (Fig. 5), as the area decreased by 13,767.40ha in 47 years.The gain in the built-up area from 1973 to 2020 is roughly equal to the loss in vegetation cover over the same period, this indicates that the built environment expanded at the expense of natural vegetation.However, there was a slight increase in the vegetation cover between 2003 and 2013, from 60.4 to 64.8% (Fig. 5) as the change in vegetation from 1973 was − 12,350.88ha in 2003 and − 10,101.96ha in 2013 (Fig. 6).This was a result of the afforestation efforts made by the city to combat the shortage of fuelwood and to fulfil the 'Garden City' vision from the 2030 Master Plan (Chingana 2013), where urban clusters would be surrounded by greenery and buffer areas (JICA 2010).
Various programmes were launched from 2009 to 2014, where indigenous tree seedlings were planted in public areas and on river floodplains (Chingana 2013).However, vegetation cover decreased again from 2013 to 2020 as newly planted trees were neglected and died within the next few years (Wiyo et al. 2015) and land allocated for afforestation was used for other developments (Chingana 2013).Wetlands experienced their largest increase between 1984 (9.0%) and 1993 (16.6%), as the change from 1973 was 116.64 ha in 1984 and 3836.52 ha in 1993, an increase of 3719.88 ha in 10 years (Fig. 6).This increase occurred during a period that vegetation decreased greatly, from 82.3 to 68.4%, indicating that during 1984 and 1993 large amounts of precipitation had caused significant portions of the land in the city to change into marshland.The wetlands then decreased in 2013 as the overall change from 1973 was 6067.44 ha in 2003 and 2569.68 ha in 2013, meaning the area decreased by 3497.76 ha.The wetlands further decreased by 2054.88 ha in 2020 as the change from 1973 to 2020 was 514.80 ha, this may be attributed to the "dry years" of 2005, 2011, and 2014 (Tadeyo et al. 2020).
Figure 7 depicts the changes in land use/cover between successive years.Each 10-year interval experienced a different amount of change in the built-up area as urban growth rates varied between the intervals, implying that certain factors had a greater impact on the urban expansion of the city in certain years.Between 1973 and 1984, the built-up area grew by 50.6%, representing an increase of 1429.20 ha due to the migration of people and government offices into Lilongwe.In the next decade, the built environment increased by 73.2%, the largest percentage change of the study period.This exponential growth by 3111.12 ha could be explained by the increased flow of investment into the city and the clearing of vegetation to build more residential and commercial buildings.From 1993 to 2003, the change in the built-up area was not as large as in the previous decade, as it only increased by 1729.80 ha, making the percentage change drop to 23.5% (Fig. 7).Between 2003 and 2013, the percentage change dropped further to 13.7% as the area increased by 1248.84 ha, the lowest change during the study period.These values are comparatively lower than those observed in the previous years, showing that the factors that contribute to urban sprawl were not of great significance during this period.One such factor may be the slow economic growth of the country during this period, as the GDP per capita remained at US$160 and the national growth rate dropped from + 5% in 1997 to − 4% in 2001 (Frankenberger et al. 2003).
The low urban growth was also a result of many economic obstacles from 2009, as there were shortages of foreign direct investment and cuts in donor aid (Cammack and Kelsall 2011).Limited cash flow and lower incomes prevented civil authorities from investing in urban development and inhibited people from building homes.However, the percentage change rose again to 54.3% from 2013 to 2020 (Fig. 7), since the built-up area expanded by 5618.16 ha.This may be the result of the increase in the city's population estimates, from 831,000 in 2013 to 1,122,000 in 2020 (UN 2019), which led to settlements expanding to accommodate rapid urbanization (Gondwe et al. 2018).From 1973 to 2020, the built-up area expanded by 465.4%, signifying that by 2020 the built-up area had expanded 5 times its size in 1973.The variations in the actual increase and percentage change of built-up areas from 1973 to 2020, can be further explained by the population estimates of the city from the United Nations population division and the population growth in informal settlements.Figure 8 shows the total city population from 1973 to 2020 (UN 2019) and includes the growth of informal settlements from 1987 to 2005 (UN 2011).Unfortunately, informal settlement population data covering the entire study period was not available, this is a limitation of our study.
Figure 8 shows that from 1973 to 1984, the population of Lilongwe City increased at a moderate rate and the first record of the informal population in 1987 was relatively small at 82,180 (UN 2011), while the estimate of the total city population was 220,000 (UN 2019).There was no need for people to settle in marginal areas while they were migrating into the newly established Capital before 1987 because between 1968 and 1984, the Capital City Development Corporation, a division of the Malawi Housing Corporation (MHC), supplied 10,400 plots of land in Traditional Housing Areas (THAs) as reported by Englund (2002).This could explain the change in the built-up area during 1973 and 1984 by 50.6%.The rate of population growth increased greatly between 1987 and 1995 as more people migrated into the city.However, due to the MHC assigning the provision of plots to the Lilongwe City Council, between 1984 and 1992, only 860 plots were provided (Englund 2002).This led to congestion in THAs, and with the need for more space, residents of the city and new migrants were drawn to informal areas (Englund 2002), increasing the informal population size from 82,180 in 198782,180 in to 141,201 in 199582,180 in (UN 2011).The rapid city population growth from 220,000 residents in 1987 to 362,000 in 1995 (UN 2019), coupled with the increase of people building homes in marginal areas may have caused the built-up area to expand by 73.2% between 1984 and 1993.With civil authorities no longer providing plots of land in designated residential areas and the high cost of building in THAs, more people occupied informal settlements, which led to the exponential growth in the informal population size from 1995 to 2005.
The informal population grew from 141,201 in 1995 to 197,041 in 2000 and more rapidly to 277,762 in 2005, an increase of 136,561, as there was limited enforcement and regulation of urban development policies.During the same period (1995)(1996)(1997)(1998)(1999)(2000)(2001)(2002)(2003)(2004)(2005), the total city population size grew by 228,000 (UN 2019), which indicates that the majority of new migrants into the city settled in informal settlements.This increase in the size of the informal population, which results in the expansion of informal settlements, contradicts the drop in the percentage change of the built-up area from 1993 to 2003 and even to 2013.Therefore, this implies that the decrease in the rate of urban expansion was due to the slow economic growth of the country during this period.Despite the informal population growing and constructing homes, the increase in urban land area was not comparable to the increase that would have occurred with rapid economic growth.Between 2013 and 2020 the population size of the city grew exponentially.This increase in city population by 291,000 people (UN 2019) was likely the cause of the rise in urban development from 2013 to 2020 by 54.3%.
Figure 9 shows the variations of the built-up areas from 1973 to 2020 in the four quadrants of the city.There was a consistent increase in impervious surfaces in all four directions over the entire study period.In 1973, all the quadrants had roughly equal proportions of the built-up area (4.7% in the south-east, 4.9% in the north-west, 6.5% in the northeast, and 7.2% in the south-west), and by 2020, all quadrants had widely different proportions.From 1973 to 1984, the rate of urban expansion was slow, with the south-west and south-east quadrants having relatively faster rates of growth than the north-east and north-west quadrants.By 1984, the south-west had the largest proportion at 11.9%, and the north-west had the smallest at 6.6%.This is similar to the results depicted in Fig. 4, where the south-west quadrant had the largest area of built-up area in 1984 and the northwest had a smaller area.But from 1984 to 1993, the rate of expansion increased, as depicted in Fig. 9.However, this increased rate occurred in only three quadrants, particularly the south-west which rose to 21.2% due to the expansion of the Old Town sector and residential areas in the quadrant, while the south-east maintained a constant rate of growth.
Between 1993 and 2003, the rate of urban expansion had declined, especially in the north-east and south-west quadrants.The north-east quadrant even experienced a drop in the proportion of built-up area in 2003 to 10.9%, as wetlands expanded during this period (Tadeyo et al. 2020).The decline also contradicted the exponential growth rate of the south-west in the previous decades.However, this decline in the rate of expansion is likely due to the slow economic growth of the country from 2001 (Cammack and Kelsall 2011).Therefore, south-west quadrant was the most affected by the limited cash flow and cuts in donor aid to the city (Cammack and Kelsall 2011), as the north-west and southeast quadrants still maintained constant rates of expansion.
The economic decline continued to affect the south-west in the next decade, as very little urban expansion occurred from 2003 (24.4%) to 2013 (24.6%).The economic decline also began to affect the north-west quadrant as the rate of growth decreased.The built environment in the north-east and south-east quadrants was unaffected and expanded to 15.5% and 20.0% in 2013, respectively.However, from 2013 to 2020 there was rapid urban sprawl as all four quadrants had increased growth rates in the built-up area.The southwest had the largest proportion of built-up area at 38.3%, the south-east with 35.3%, and the north-west with 31.9%.It is worth noting that high-to-medium-density suburbs, where houses are constantly constructed, are found in these quadrants.The north-east quadrant had the least urban expansion at 26.3%, which is likely due to low-density suburbs, and the Kanengo industry sector due to strict enforcement of urban development regulations.As the built-up area in each quadrant occupies less than 40% of the total land area of each quadrant, this implies that the city is not saturated with impervious surfaces and that natural vegetation still covers most of the sections of the city.However, it can be expected that urban land will reach 50% as the 2030 Master Plan sets out that half of the city's area be converted to the built environment (residential, commercial, and industrial purposes) and that low and middle-density areas be densified, while the other half remains vegetated land and open spaces (JICA 2010).

Shannon's entropy metrics
Figure 10 shows that the city had undergone a rapid dispersed urban sprawl, as the values of relative Shannon's entropy indices from 1973 to 2020 are above 0.5 and close to 1.The relative Shannon's entropy indices (Fig. 11), implies that urban sprawl was not consistent over the entire study period.The growth of the built-up area was increasingly fractured from 1973 to 1993, as the relative entropy index increased from 0.903 in 1973 to 0.941 in 1993.However, the rate of urban sprawl decreased from 1993 to 2013 as illustrated by the negative relative Shannon's  11).The low relative Shannon's entropy indices may be a result of infill development and the expansion of existing urban structures, leading to a more stabilized urban growth between 2003 and 2013.However, the relative entropy index increased again by 2020 to 0.966, the highest of this study period.This implies that the rate of urban sprawl increased and that this new growth was fractured and dispersed.
The variations of relative Shannon's entropy indices in the four quadrants from 1973 to 2020 are shown in Fig. 12.In 1973, the relative Shannon's entropy indices in the four directions were extremely different, with the north-east having the highest index of 0.925, hence the most fractured urban landscape, and the south-west having the lowest index of 0.770, meaning the urban landscape in this quadrant was the most compact.Although Fig. 9 shows that the built-up area in each quadrant was roughly similar in 1973, Fig. 12 illustrates that the built-up area in each quadrant differed in the degree of dispersion.Between 1973 and 1984, the relative Shannon's entropy index had increased, implying that urban growth was becoming more irregular and dispersed.However, the north-east quadrant experienced a decrease to 0.904 in 1984 (Fig. 12).Between 1993 and 2013, there was a noticeable decline in the relative Shannon's entropy, with the south-east and south-west only having a slight increase.This is supported in Fig. 11, which shows that the relative entropy declined between 1993 and 2013.Finally, from 2013 to 2020, the relative entropy index increased in all directions, with the south-west having the highest index of 0.975, the northeast having the second-highest index of 0.965, south-east with 0.953, and the north-west having the lowest of 0.932 (Fig. 12).The high relative entropy index for the northeast quadrant contradicts the low proportion of built-up land (26.3%), while the high relative entropy index for the south-west supports the high proportion of built-up land (38.3%) as shown in Fig. 9.This indicates that though the north-east quadrant consists of formal and low-density suburbs, new developments are occurring in a dispersed manner, while other quadrants like the north-west with the most development (31.9%) are more compact.

Built-up area forecast
Regression analysis revealed that the regression line fits the input data closely, as R 2 was equal to 0.956.This meant that 95.6% of the variation in the built-up area could be explained through the variations in the population size.The relationship between the independent and dependent variable was statistically significant as the p value of the independent variable, population size, was less than 0.001.The resulting regression line from the analysis is expressed by Eq. 6.
where y , the dependent variable, is the area of the built envi- ronment in hectares in a particular year and x , the inde- pendent variable, is the population of Lilongwe City in that particular year.Table 1 presents the predicted values and the residual errors from the observed built-up area.
The predicted values exhibit a slow growth in the urban land area between 1973 and 1984 just as the observed values.The rate of urban expansion rose from 1984 to 2003 with the built-up area increasing by approximately 2000 ha between each 10-year interval.However, between 1984 and 1993, the predicted values did not reflect the increased growth rate of the built-up area, as the values increased by less than 2000 ha while the observed values increased by 3111.12 ha.This rise in the expansion rate of the built environment during this decade was caused by population growth and the increase in foreign direct investment into the city (Cammack and Kelsall 2011).Therefore, without considering the economic growth of the country in the analysis, the residual of 1993 is relatively large.The rate of urban expansion in the predicted values then increased between 2003 and 2020, with an increase of approximately 3200 ha between each 10-year interval but was contrary to the slow growth in the observed built-up area from 2003 and 2013 as it only increased by 1248.84 ha.Just as the economy affected (6) y = 2609.28+ 0.0112x  The rate of increase in the predicted values also did not mirror the exponential rise between 2013 and 2020, where the built-up area increased by 5618.16 ha.Therefore, the residuals for 2013 and 2020 are relatively large as the regression model did not include other variables such as economic data that may have contributed to the decline in urban expansion between 2003 and 2013 and the increase between 2013 and 2020.Figure 13 displays the relationship between the builtup area and population size with the fitted regression line, from 1973 to 2020.The built-up area and population were positively correlated.This relationship allowed the built-up area in 2023 and 2033 to be predicted by substituting the population projections into Eq.6, as shown in Table 2.The estimates of population display a gradual increase from 1973 to 1993 and eventually a more rapid increase from 1993 to 2020, as shown in Table 2.The population projections indicate that the city will have over a million people by 2023 and that this population size will nearly double by 2033.By 2023, the built-up area of Lilongwe City is projected to increase to 16,954.45ha, and 10 years later (2033) the built-up area is projected to reach 25,228.78ha.This indicates that the following decade will experience exponential growth, as the built-up area will expand by approximately 8300 ha, the largest 10-year increase.The rapid change in the built-up area is a direct result of the projected 736,000 increase in population from 2023 to 2033.It is worth noting that these estimates may be affected by prevailing economic conditions in Lilongwe City and the entire country (Malawi) in the next decade and beyond.

Conclusion
The current study was carried out to determine and analyse the spatial and temporal expansion of Lilongwe City from 1973 to 2020 using semi-dynamic land use/cover analysis, regression analysis and Shannon's entropy modelling.Landsat images from 1973 to 2020 in a nearly 10-year interval were used to determine spatial-temporal land use/ cover changes while Shannon's entropy model was applied in modelling the degree of dispersity of the city's urban sprawl.Finally, a linear regression model incorporating population as an explanatory variable was used to predict future spatial expansion trends for Lilongwe City.The spatial urban growth trends were determined by zoning the city into concentric rings and quadrants.Results revealed that the built-up area in Lilongwe City expanded at a rate of + 9.9% per year while vegetation cover decreased at a rate of -0.7% per year between 1973 and 2020.High relative Shannon's entropy indices (0.9 and above) were observed, indicating a dispersed urban development pattern.The spatial trends showed that urban growth was prominent in the south-east, south-west, and north-west quadrants and that the quadrant with the least urban development (north-east) had the most dispersed urban development.Linear regression analysis indicated exponential urban growth for Lilongwe City in the next decade due to the projected high rate of population growth.
By investigating the spatial-temporal dimensions of the urban expansion of Lilongwe City, the factors that inhibited and promoted growth in certain periods were identified, such as the expansion of wetlands in the eastern and most northwestern parts of the city, the dependence on urban agriculture as a source of income, rural-urban migration, innerurban congestion in the city due to the inconsistent provision of housing in the Traditional Housing Areas (THAs), the absence of land-use enforcement, the economic growth of the country, and the expansion of industrial activities in the Kanengo sector and along the Nacala corridor.The correlations between these factors and spatial-temporal urban expansion would enable civil authorities to formulate policies to control urban expansion and ensure growth occurs according to the 2030 Master Plan and other urban development policies.
. It is worth noting that the 1984 image was used instead of the 1983 image because there was no clear Landsat image for the area of study in 1983.The 1973 image was captured by Landsat 1-5 MSS C1 Level-1 sensor satellite.The 1984 and 1993 images were captured by Landsat 4-5 TM C1 Level-1 satellite sensor.The 2003 image was captured by Landsat 7

Fig. 1
Fig. 1 Location map of Lilongwe City

Fig. 4
Fig. 4 Temporal variations in land use/cover in the four quadrants over Lilongwe City from 1973 to 2020

Fig. 8
Fig. 8 Informal settlement population and the total population of Lilongwe City from 1973 to 2020

Fig. 9
Fig. 9 Variations of the built-up area in each zone over Lilongwe City from 1973 to 2020

Fig. 12
Fig. 12 Directional relative Shannon's entropy index in four principal directions for Lilongwe City from 1973 to 2020

Fig. 13
Fig. 13 Modelled relationship between the built-up area and population in Lilongwe City

Table 1
Observed and predicted built-up area by regression analysis (units are in ha)

Table 2
Predicted built-up area for 2023 and 2033 based on population estimates in Lilongwe City