Measuring Vertical Urban Growth of Patna Urban Agglomeration Using Persistent Scatterer Interferometry SAR (PSInSAR) Remote Sensing

Abstract

In the present study, the vertical and horizontal growth of Patna Urban Agglomeration was evaluated using the Persistent Scatterer Interferometry Synthetic Aperture Radar (PSInSAR) technique during 2015–2018. The vertical urban growth assessment of the city landscape was assessed using microwave time series (30 temporal) datasets of Single Look Complex (SLC) Sentinel-1A interferometric Synthetic Aperture Radar using SARPROZ software (ver. 2020). This study demonstrated that peripheral city regions experienced higher vertical growth (~4 m year⁻¹) compared to the city core regions, owing to higher urban development opportunities leading to significant land use alterations, the development of high-rise buildings, and infrastructural development. While the city core of Patna observed an infill and densification process, as it was already saturated and highly densified. The rapidly urbanizing city in the developing region witnessed a considerable horizontal urban expansion as estimated through the normalized difference index for built-up areas (NDI_B) and speckle divergence (SD) using optical Sentinel 2A and microwave Sentinel-1A ground range detected (GRD) satellite data, respectively. The speckle divergence-based method exhibited high urban growth (net growth of 11.28 km²) with moderate urban infill during 2015–2018 and reported a higher accuracy as compared to NDI_B. This study highlights the application of SAR remote sensing for precise urban area delineation and temporal monitoring of urban growth considering horizontal and vertical expansion through processing a long series of InSAR datasets that provide valuable information for informed decision-making and support the development of sustainable and resilient cities.

1. Introduction

The accelerated urban growth in developing countries has led to a significant alteration in the physical and functional properties of the landscape [1]. Moreover, the 2D representation of urban landscape and its complex intrinsic processes, including urban growth, seldom presents the true composition of urban morphology and landscape [2]. However, with the growing concentration of the global population (~54%) within ~3% of the earth’s landmass [3], it is necessary to understand the complete dimensions of urban growth in horizontal and vertical spaces. The use of remote sensing for mapping urban landscapes and growth processes at varied times and scales is not very new. Remotely sensed satellite images are being widely used to analyze the impact of anthropological activities on the natural landscape, analyze urban footprints on the environment, quantify climatic variability, and provide suitable urban and regional planning processes [4]. Various studies used various multi-resolution satellite images for the chronological evolution and development of a city [5,6], the characterization of essential urban landscape attributes [7], urban heterogeneity [8], and other parameters of changing human–environment relationships in spatio-temporal contexts [9,10].

The recent advancements in the field of remote sensing and its attributing technologies have provided a lot of precise tools for the extraction of different features from satellite images. Scholars have integrated very high-resolution satellite imagery with the neural network-based deep learning method, along with the guided filter to extract building pixels [11]. Additionally, a combination of high-resolution panchromatic imagery with unsupervised clustering based on histogram peaks and edge detection using a Canny operator was used to extract the building. The use of very high-resolution images (such as Landsat 8, SPOT 5, and Sentinel 1 and 2) can improve the detection of small changes, especially considering that pixel level changes are often omitted by the automatic classification [12,13,14]. Later, PCA-based normalized index-based built-up area extraction was also reported using a simple method for built-up area extraction using the unique spectral bands of WorldView-II satellite images reported for urban area extraction and density analysis [15]. A normalized difference index (NDI) is a dimensionless index that describes the difference between visible (blue) and near-infrared reflectance of built-up cover and can be used to estimate the density of built-up areas. The extent of the urban area is also analyzed with the help of a normalized difference index, which was originally proposed by Rouse et al. (1973) for the identification of vegetated areas. One of the commonly used NDIs is the normalized difference built-up index (NDBI), which is based on the principle of analyzing the spectral characteristics of satellite imagery and calculating the normalized difference between near-infrared (NIR) and shortwave infrared (SWIR) bands and measures the extent of urban growth or built-up areas of a specific region [16,17]. The index emphasizes the presence of built infrastructure, as urban areas often have higher reflectance in the SWIR region in contrast to natural elements such as vegetation or water bodies [18]. Using NDBI, in combination with other normalized indices, this article assesses the impact of growing urbanization on land surface temperature, emphasizes the urban heat island effect, and discusses the significance of NDBI for investigating the environmental effects of urbanization [19].

Urban planners have long debated the advantages of vertical urban development (the concentration of people via the construction of tall structures) over horizontal urban expansion (the dispersion of population through the extension of urban limits). The possibility and consequent urgency of these effects as a result of cities’ vertical expansion have, however, only been evaluated in a small number of studies [20]. By conserving the city’s peripherals for wildlife or agriculture, urban densification through vertical rise is also recommended to support sustainability [21]. A study made use of GIS, and its 3D modeling function, to investigate the urban skyline and its transformation due to new high-rise constructions and to construct, assess, and analyze the city silhouettes. It also demonstrated how the local decision-making context affects the efficacy of these strategies for evaluating and pre-testing tall construction projects [22]. A novel general equilibrium model of vertical and horizontal city structure was constructed in a prior work to investigate the causes and consequences of vertical growth [23]. Using the probabilistic principles governing extreme values, a prior study considered tall structures as sporadic exceedances of a threshold, and this work extrapolates the characteristics of the skyscrapers that would dominate the urban skyline in 2050 [20]. Moreover, very high urban skyline development introduces many challenges, including large shadow coverage in the vicinity, overexploitation of groundwater, alteration of the urban microclimate, and upper air circulation.

Although optical remote sensing has a wide optical wavelength region that has been widely used in urban growth mapping and monitoring on horizontal surfaces, it has limitations in providing growth in the vertical dimension. A comparative study was performed using the best pair of bands for normalized differencing to extract built-up areas [24]. With the advent of synthetic aperture radar (SAR), the scope of urban area monitoring has been tremendously increased due to its sensitivity to the polarimetric properties of man-made and natural scatterers [25]. Studies reported the detection of urban landscapes using speckle divergence analysis, which efficiently highlights areas characterized by heterogeneous and highly structured built-up sections [26,27]. Using chain-based Gaussian models and fuzzy K-means unsupervised classification, the potential of SAR and optical data combined has been leveraged for quick urban mapping [28]. The KTH-Pavia Urban Extractor, coupled with spatial indices and grey-level co-occurrence matrix textures, was used to achieve worldwide urban mapping using SAR data [29]. The Persistent Scatterer Interferometry Synthetic Aperture Radar (PSInSAR) approach is a more sophisticated differential interferometry synthetic aperture radar (DInSAR) technology for detecting and monitoring the Earth’s surface movement over time. It combines multiple synthetic aperture radar (SAR) images acquired over time to detect and measure subtle surface movements (such as subsidence or uplift) at different scales (regional or local). It utilizes coherent and persistent radar reflectors known as persistent scatterers (PS), which are typically man-made structures or natural features with stable and consistent radar responses over time [30]. These scatterers, often located on building roofs or other stable surfaces, act as reference points for the analysis of the interferometric phase information contained in SAR images [31]. It addresses the constraints of the DInSAR technique’s geometric, temporal decorrelation, and atmospheric effects [32].

The main difference between the DInSAR and the PSInSAR is the number of datasets they require for processing; the former requires a comparatively smaller number than the latter [33]. The main advantage of using the PSInSAR technique over DInSAR is the computation of the differential interferograms using digital elevation models, which enhances the precision of the technique up to millimeters [34]. Generally, PS points can be detected more easily in urban areas, because of the high-rise buildings, than in rural areas, due to the presence of agriculture or forest areas which keep changing with time [35]. PSInSAR, even though it is a sound technique to detect changes very precisely, has some limitations. The prime limitation is the ability to perform temporal sampling of the deformation, which directly depends upon the temporal resolution of the SAR satellite [33]. PSInSAR is a valuable tool for vertical urban growth assessment as it provides high-resolution information [36] and accurate and precise measurements of vertical deformation, which can help in assessing the magnitude and distribution of vertical urban growth with high precision and accuracy [16]. It is also a non-destructive remote sensing technique that does not require physical access to buildings or construction sites, which makes it a cost-effective and safe method for monitoring urban growth [37]. PSInSAR allows for the detection of ground deformation in near real-time, which can be useful for monitoring the pace and direction of urban growth and for informing urban planning and management decisions. It can also provide information on urban growth patterns over a large spatial extent, making it useful for regional and city-level planning and management [38].

Understanding the dynamics of urban growth, encompassing both vertical and horizontal expansion, is crucial for comprehending the urbanization process, population concentration, socio-economic activities, and environmental impacts and thereby formulating suitable urban planning policies. This study presents a novel approach for assessing the vertical and horizontal growth of the rapidly expanding Patna City, which has undergone significant land use transformation in recent decades. Notably, the migration from rural to urban areas has played a pivotal role in driving urban growth in developing countries such as Patna. This influx of migrants has presented numerous socio-economic challenges for urban centers and their surrounding regions. In response to the growing needs of the population, there has been a surge in the construction of residential and commercial infrastructure, as well as the development of transportation networks. Additionally, the government has launched the Smart City Mission, aiming to transform Patna into a sustainable and livable urban environment through infrastructure improvements, enhanced mobility, and better public services. To address these crucial aspects of urban growth, this study focuses on the vertical and horizontal expansion of the Patna Urban Agglomeration, utilizing the innovative Persistent Scatterer Interferometry Synthetic Aperture Radar (PSInSAR) technique during the period of 2015–2018. By leveraging SAR remote sensing, this study employs precise urban area delineation and temporal monitoring of urban growth, considering both the horizontal and vertical dimensions through the processing of an extensive series of InSAR datasets. This research contributes to the field by providing a comprehensive understanding of the intricate urbanization patterns in Patna, considering the interplay between vertical and horizontal growth. The use of advanced remote sensing techniques adds a novel dimension to the analysis, enabling accurate and detailed assessments of the city’s evolving urban landscape in a near real-time manner that supports urban planners, policymakers, and researchers in effective urban growth management and fosters sustainable urban development.

2. Study Area

Patna is the state capital of Bihar state in eastern India and one of the oldest continuously inhabited places in the world and is referred to as Patna Urban Agglomeration in this study [39]. Lying at an average elevation of 53 meters above sea level, it covers an area of 127.86 km². (Figure 1). The present structure of the townscape of Patna is directly unique to its site and situation. The natural ridge of the River Ganga has provided a unique site and location for this city since 490 BCE [39]. The urban population of Patna rose four-fold between 1971 and 2011, from a mere 473,000 in 1971 to around 1.68 million in 2011 [40]. Furthermore, the city’s population growth has not been consistent. During the 1970s, this city recorded the highest growth, which decreased dramatically in the next decade [41]. The old and rapidly expanding city of Patna, the capital of Bihar, is situated along the Ganges River’s bank. The municipal area of the city is 99 km², and its adjoining suburban area is 36 km² [42]. The overall population of the Patna Urban Agglomeration (PUA) was approximately 2 million, a 23.73 percent increase from 2001 [43].

3. Data Used and Methodology

The methodology consists of the analysis of databases generated from satellite images, ancillary databases, and information collected in the field. Various optical, microwave remotely sensed data sources, including Sentinel 2A and Sentinel 1A satellites of the European Space Agency, were utilized to estimate vertical and horizontal urban growth. A total of 32 temporal microwave SLC datasets from Sentinel 1A were used to estimate vertical urban growth (

Table 1. Details of Satellite Data.

Satellite	Type	Spatial Resolution	Date of Acquisition
Sentinel 2A	Multispectral	10 m, 20 m, and 60 m	13 February 2015 and 14 December 2018
Sentinel 1A	GRD	10 m	15 January 2015 and 25 December 2018
Sentinel 1A	SLC	10 m	15 January 2015, 8 February 2015, 16 March 2015, 9 April 2015, 3 May 2015, 15 May 2015, 27 May 2015, 20 June 2015, 14 July 2015, 26 July 2015, 31 August 2015, 30 October 2015, 6 October 2015, 23 November 2015, and 17 December 2015 11 May 2018, 23 May 2018, 4 June 2018, 16 June 2018, 10 July 2018, 22 July 2018, 3 August 2018, 27 August 2018, 8 September 2018, 20 September 2018, 2 October 2018, 26 October 2018, 7 November 2018, 19 November 2018, and 1 December 2018

). The used microwave (SLC) time-series datasets were acquired by Sentinel-1A in an ascending pass track with an incidence angle (in degrees) of 38.3 and a maximum perpendicular baseline of 700. Different software, viz., ArcMap 10.4, ESA SNAP, Q-GIS, Google Earth, and SARProZ, were used to process the satellite data and its analysis.

3.1. Horizontal Urban Growth Assessment

For assessing horizontal urban growth, a microwave ground range detected (GRD) dataset (Sentinel-1A), based on the speckle divergence method, and an optical dataset (Sentinel 2A), based on the normalized difference index for built-up areas (NDI_B), were used to extract built-up features.

3.1.1. SAR-Based Built-Up Area Extraction Using Speckle Divergence

The horizontal growth in Patna city was estimated using texture feature-based speckle divergence analysis from January 2015 to September 2018 by characterizing heterogeneous and highly structured built-up areas [26,27]. The high values of speckle divergence highlight the settlement areas. The double bounce effect causes strong scattering in urban areas from SAR data; hence, the local speckle is focused on the analysis of local speckles. The large segments of the image focused on the rough delineation of settlement areas; the settlement bodies are represented as the few image objects. The result of the speckle divergence analysis demonstrated the potential of texture extraction efficiency. It was a very efficient method based on the sigma probability of the Gaussian distribution of speckle noise and preserved true structures with texture and contour information [27]. The authors employed a technique called “speckle divergence analysis” to analyze local statistics using a straightforward approach. Their method suppressed noise more effectively than traditional filtering methods, which typically retain the true structures along with texture and contour information [27]. The filter utilized in this analysis was based on the sigma probability derived from the Gaussian distribution of speckle noise. It successfully removed speckles and reduced noise while preserving important texture, structure, and contour details [44]. This aspect was crucial as the study area consisted of numerous regions characterized by small-scale structures. The resulting speckle divergence image was then utilized for object-based settlement classification. The classification results were satisfactory and displayed good accuracy. Implementing this method to analyze the study area in this research holds promise for producing improved outcomes. The speckle was removed, and the noise was suppressed, but the texture, structure, and contour information were well preserved. The speckle divergence image was used for object-based settlement classification. The calculation of speckle divergence was based on the following equation deduced by Esch et al. (2010):

$$D_{x,y}=C_{x,y}-C\mathrm{where},C_{x,y}=\frac{{\sigma }_{x,y}}{{\mu }_{x,y}}$$

(1)

D_x,y: the speckle divergence.

C: the theoretical heterogeneity due to developed speckles; it was calculated from the inverse of the equivalent number of looks.

C_x,y: the local coefficient of variation defined by the local.

σ_x,y : the local standard deviation.

μ_x,y: the local mean.

The speckle divergence ranges between 760 and 4661 and 1114 and 2799 for the years of 2015 and 2018, respectively. The thresholding of 1295.431 to 1739.074 was applied on S1A SAR images for 2015 and 2018 to demarcate the urban area, mainly in the range of 1497.254 to 1834.254.

3.1.2. Optical-Based Built-Up Area Extraction Using the Normalized Difference Index

The normalized difference built-up index [45] was applied to extract built-up land using the blue and near-infrared bands of the Sentinel-2A multispectral satellite data during 2015-2018. The horizontal growth was assessed by comparing the results obtained by optical remote sensing-based speckle divergence analysis over the different datasets (15 January 2015 and 25 December 2018). The horizontal growth of Patna City was assessed by analyzing temporal changes in the urban area over the years, using the normalized difference index for built-up areas.

$${\mathrm{NDI}}_{\mathrm{B}}=\frac{\mathrm{Blue}-\mathrm{NIR}}{\mathrm{Blue}+\mathrm{NIR}}$$

(2)

where NDI_B—the normalized difference index for built-up areas, NIR—the near-infrared band, and Blue—blue band.

The result from the blue band and near-infrared band was very good for the extraction of the built-up area when compared to other combinations [24].

3.2. Vertical Urban Growth Assessment

Persistent Scatterer Interferometry is a sophisticated differential interferometry synthetic aperture radar technology that can calculate and track the Earth’s surface movement across time. This includes the master image collection process, co-registration of a single master image, generation of interferograms and differential interferograms, phase unwrapping, selection of PS targets, elimination of atmospheric phase, the study of time series, and measurement of displacement [32]. The identification of stable, secure targets called persistent scatterers is the main point of this technique. PSs are stable pixels that are found in all chosen images with the same scattering properties over time, e.g., houses, pipelines, electrical poles, or artificially mounted corner reflectors [35]. The final persistent scatterers are a subset of the persistent scatterer candidates after estimating the phase stability of the pixels. The amplitude stable pixels are calculated using the Amplitude Dispersion Index [38,46,47].

$${\mathrm{D}}_{\mathrm{A}}=\frac{{\sigma }_A}{{\mu }_A}$$

(3)

where, ${\sigma }_A$ is the temporal standard deviation of each pixel, and ${\mu }_A$ is the temporal mean of each pixel.

The interferometric phases of the persistent scatterer candidate (x) in ith interferogram, can be written as [47,48]:

ϕ_int,x,i = W{ϕ_def,x,i + ϕ_a,x,i + ϕ_orb,x,i + ϕ_ε,x,i + n_x,i}

(4)

where ϕ_def,x,i is the surface deformation phase component, ϕ_a,x,i is the atmospheric phase delay component, ϕ_orb,x,i is the phase component due to orbital errors, ϕ_ε,x,i is the look angle error phase component, and n_x,i is the noise. W is the wrapped operator. A band-pass filter was used to estimate the correlated phase contributions due to the surface deformation, atmosphere, and orbit. The temporal coherence ($\gamma$) of all persistent scatterer candidates, after correcting the spatially correlated phase components and the look angle error, was calculated to estimate the phase noise level [48,49].

$${\gamma }_x=\frac{1}{N}\left|{{\displaystyle \sum }}_{i=1}^{N}exp\left\{\varphi int,x,i-\underset{\_}{\varphi }int,x,i-\Delta \widehat{\varphi }\epsilon ,x,i\right\}\right|$$

(5)

The above equation calculates the temporal coherence of a persistent scatterer candidate (x), where N represents the number of interferograms, ϕ_int,x,i is the estimated spatially correlated components, and $\Delta \widehat{\varphi }\epsilon ,x,i$ is the estimated look angle error. The persistent scatterer candidate with high temporal coherence (${\gamma }_x$) was selected as the new persistent scatterer candidate, and the remaining candidates were rejected. The four-phase components of surface deformation, atmosphere, orbit, and look angle were then reassessed for this selected persistent scatterer candidate, and the process was iterated repetitively.

Accuracy assessment was performed by calculating the kappa coefficient for overall accuracy with the help of 100 random ground control points taken through Google Earth [50]. The accuracy assessment of the classified images indicated that the Kappa coefficient value for speckle divergence ranged from 0.81 to 0.89 for 2015 and 2018, while the Kappa coefficient for the normalized difference index for built-up area extraction ranged from 0.79 to 0.84. To validate the results, some important landmarks and static locations (vertical growths that are neither the construction nor destruction of storeys) in Patna Urban Agglomeration were selected for validation. The vertical growth for these permanent locations was studied, and it was observed that the majority of the city has undergone insignificant vertical growth from January 2015 to December 2018, barring a few locations.

4. Results and Discussion

4.1. Horizontal Growth of Patna Urban Agglomeration

The built-up footprint of Patna Urban Agglomeration was mapped using (a) Sentinel 2a (multispectral)-based normalized difference index for built-up areas (NDI_B), and (b) Sentinel 1A/B SAR-based speckle divergence methods during 2015–2018 (Figure 2). This study provides a comparative evaluation of SAR-based speckle divergence and multispectral Sentinel-2A data, specifically employing the NDI_B for the built-up footprint. Focusing on Patna Urban Agglomeration during the period from 2015 to 2018, characterized by urban expansion, the SAR-based speckle divergence method leveraged the analysis of speckle patterns within SAR images acquired at different temporal intervals. This approach yielded a built-up area growth from 63.33 sq. km to 74.6 sq. km during 2015–2018, with 17.8% growth (Figure 2c,f). In contrast, the multispectral Sentinel-2A data approach employed the NDI_B to detect the built-up area and its growth, exhibiting the built-up area growth from 69.94 sq. km to 76.45 sq. km during 2015–2018 with 9.31% growth (Figure 2b,e and

Table 2. Built-up and non-built-up areas derived from different methods for 2015 and 2018.

	Method for Built-up Area Extraction
	Speckle Divergence		Normalized Difference Index
Area of Features (in km2)	2015	2018	2015	2018
Built-up	63.33	74.6	69.94	76.45
Non-Built-up	64.537	53.267	57.927	51.417

). The NDI_B-based study to distinguish urban built-up areas is entirely based on the spectral reflectance of the land use/ cover, though it is more clearly distinguishable in 2018 as compared to the 2015 observation. However, the speckle divergence analysis mainly involves the identification of features by giving even bounce scattering, especially double bounce scattering (Figure 2c,f), which the built-up features dominantly provide [51]. Thus, in the present study, having a double bounce speckle divergence was categorized as a built-up area feature. However, the normalized difference index works with reflectance, which is far more exact than scattering in the case of speckle divergence (Figure 2b,e). Comparing the two methodologies, it was observed that both methods effectively captured urban expansion, albeit with slight disparities in the estimated values for the built-up footprint. Notably, SAR data, characterized by its all-weather imaging capability, insensitivity to atmospheric interferences, and sensitivity to built-up area structure, offers significant advantages in effectively mapping built-up areas, particularly in areas prone to cloud cover. By exploiting backscatter properties, this technique provides valuable insights into the structural characteristics of urban environments, thereby enabling precise urban planning and management.

This study exhibited maximum horizontal urban growth in the urban fringe, primarily in the southern parts of Patna, while the central parts of the city turned out to be denser due to infill processes. The Patna Metropolitan Planning Committee’s “Patna Master Plan 2031” states that the city’s urban area has been increasing significantly, especially in its periphery [52]. The urban growth analysis of Patna’s urban agglomeration indicated that the city has mainly expanded in the southern and eastern directions (Figure 2). The built-up area expanded more southwards to Jethuli (including Kumhrar to Pahari village, Ramakrishna Nagar to Kacchuara, Davnandan Kunj to Doman Chak, and Sipara to Pakri) in 2018 compared with the built-up area distribution during 2015, which was only till Khizirpur. Moreover, the eastern part of the city observed slight variations, whereas the western and northern parts observed insignificant changes. This may be attributed to the location of the holy river Ganga in the north and the cantonment area in the west, which are major constraints for urban growth in the respective directions. A recent study on the Patna municipal corporation indicated that the expansion of Patna city has been mostly towards the southern, north-eastern, and north-western parts during 2012–2022, while the existence of the Ganga in the north restricts built-up area expansion in this particular direction [47].

4.2. Vertical Growth of Patna Urban Agglomeration Using PSInSAR

The vertical growth of Patna Urban Agglomeration was mapped using the PSInSAR method employing 32 SLC datasets of Sentinel-1A (VV polarization) in SARProZ^TM (a purpose-based application) software during 2015–2018 (Figure 3). The PSInSAR developed by Ferretti et al. (2000) [53] is based on a conventional radar and is a new technique to determine surface displacement. The study area, strongly affected by temporal decorrelation, has observed both horizontal and vertical urban growth during the observation period (2015–2018). The present study utilized 32 Sentinel-1A (spatial resolution: 10 m) SAR images for the entire study area (with a maximum relative temporal baseline), which were co-registered on a unique master (Sentinel-1A orbit taken in May 2015). The multi-temporal vertical urban growth study indicated high vertical growth (3 to 4 m/year) in central Patna, the southern outskirts of Pahari locality, Kachhuara, Pakri, and some growth in the eastern outskirts of Patna city, comprising Sabalpur, Kothiya and Jethuli. Moderate vertical growth rates (2 to 3 m/year) were observed in Ramakrishna Nagar, Kumharar, and Sheikhpura in the north, and the locations that observed moderate vertical urban growth included Patliputra colony in the north, Khajpura in the west, and Mahesh Nagar and Jay Prakash Nagar the central region. In contrast, P.C. Colony, Tilak Nagar, and Khizirpur in the east exhibited low vertical urban growth rates (>2 m/year). Vertical urban growth was characterized by the increase in the number of storeys of already existing buildings, the construction of new buildings, and the destruction of old and existing buildings, which were recorded as a decrease in vertical growth. The 6 tandem pairs after DEM compensation were used to generate 31 differential interferograms, which were resampled on the uniform image grid by Kriging interpolation. The resultant PSInSAR-based vertical growth of Patna Urban Agglomeration was overlaid on Google Earth to comprehend the trend of vertical growth in the parts of the city region (Figure 4).

PSInSAR allows for the detection and measurement of ground deformation over time, which can be attributed to various factors, including urbanization. Several studies have demonstrated the effectiveness of PSInSAR in detecting vertical urban growth. Previously, the vertical deformation of buildings was monitored in Shanghai, China, and found a significant increase in building height over the past decade [16]. Another study more accurately assessed the vertical urban growth in Kolkata City (India) using PSInSAR and reported a high rate of vertical deformation in the central business district, indicating a concentration of high-rise buildings in this area and identifying zones of potential future growth [37]. The analysis of the results has been performed by observing the areas classified as having the maximum vertical growth with the aid of Google Earth. Some real-time field photos from both 2015 and 2018 from Google Earth were utilized for validation of this study and exhibited high vertical growth together with excessive densification and infill in the central and eastern parts of the city (Figure 5).

The findings of this study were supported by the discernible pattern of outward horizontal expansion of the urban area towards its periphery in Patna. This spatial phenomenon is characterized by the proliferation of new built-up features on previously unoccupied land, demonstrating the process of urban sprawl. Notably, this horizontal growth is strongly correlated with a concurrent acceleration in vertical growth rates, indicative of intensified urban development. The observed relationship between horizontal and vertical growth signifies the multifaceted nature of urban expansion, encompassing both lateral expansion and the construction of taller structures within the built-up areas (Figure 6). Such empirical evidence underscores the complex dynamics of urbanization and highlights the need for comprehensive spatial analysis techniques that account for both horizontal and vertical dimensions when assessing urban growth and its impact on land use dynamics.

This study highlights several significant advantages of temporal SAR images, and more specifically PSInSAR images, over optical satellite datasets for urban mapping. However, like any remote sensing technique, PSInSAR has limitations that need to be considered while interpreting the results for vertical urban growth assessment. The PSInSAR dataset may have a limited vertical resolution, especially in urban areas with complex building structures, which can affect the accuracy of the measurements [16]. The vegetation can interfere with the radar signal, resulting in errors in the measurements [37]. PSInSAR measurements are also limited by the available satellite data, which can result in gaps in the temporal coverage [38]. Moreover, the acquisition and processing of PSInSAR data can be expensive, which may limit its application in some contexts [37]. However, there are certain measures to overcome these limitations, which involve ground truthing to validate PSInSAR data. To enhance the accuracy and interpretation of PSInSAR, optical datasets or LiDAR data can be integrated, which can provide complementary information about land use, land cover, and land surface characteristics [54,55]. Applying advanced processing techniques can help overcome limitations in PSInSAR data. These techniques may include phase unwrapping algorithms, filtering methods, and noise reduction approaches [38,56]. Optimizing the data acquisition strategy can help overcome limitations related to the availability and coverage of PSInSAR data. This may involve revisiting the area of interest more frequently, considering different satellite orbits, or acquiring data from multiple SAR sensors. Increasing the temporal and spatial coverage of PSInSAR data can improve the accuracy and resolution of deformation measurements [57,58].

Future research can be focused on developing algorithms for identifying and monitoring potential regions of instability or structural deformation. This would allow for proactive maintenance and the prevention of structural failure, and ensure the stability of urban infrastructures (including buildings, bridges, and tunnels), using advanced field methods and PSInSAR techniques. The effects of urbanization on natural resources (viz., groundwater levels and vegetation health) can effectively be analyzed using the vertical growth of the city landscape. Moreover, the role of the vertical rise of skyscrapers in regulating the urban climate, heat stress, and alteration in local climate conditions are a few of the major challenges to investigate using PSInSAR. Further, fusing PSInSAR with advanced remote sensing technologies such as LiDAR can provide a more detailed analysis of urban structures and their vertical movements. Future research on measuring vertical urban growth using PSInSAR holds significant potential for urban planning, infrastructure management, and environmental monitoring. By advancing analytical techniques, integrating data from various sources, and exploring new applications, researchers can contribute to more sustainable and resilient urban development.

5. Conclusions

The present study highlights the use of SAR satellite data for mapping the urban growth of Patna Urban Agglomerations at horizontal and vertical dimensions using the speckle divergence method and the PSInSAR technique, respectively. This study observed moderate horizontal urban growth with an infill process and high vertical growth (~4 m/y) in parts of the Patna urban region. The observed relationship between horizontal and vertical growth signifies the multifaceted nature of urban growth, encompassing both lateral expansion and the construction of taller structures within the built-up areas. This study reported the higher accuracy of SAR-based speckle divergence compared to multispectral-based normalized different indexes for built-up area delineation. Such empirical evidence underscores the complex dynamics of urbanization and highlights the need for comprehensive spatial analysis techniques that account for both horizontal and vertical dimensions while assessing urban growth and its impact on urban ecosystems. Moreover, the role of the vertical rise of skyscrapers in regulating the urban climate, heat stress, and alteration in local climate conditions are a few of the major challenges to investigate using PSInSAR in the future. The effects of urbanization on natural resources, human health, and biodiversity can effectively be analyzed using a periodic, appropriate assessment of the horizontal–vertical growth of the city landscape.