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We provide a summary of the DSM but direct readers to the published literature for more mathematical and algorithmic details (Nghiem et al. 2009). First, QuikSCAT egg data are processed by a fast Fourier transform accounting for the Doppler compensation to generate a sub‐footprint thin “slice” data with a resolution of 6 km in range and 25 km in azimuth. For a given location on Earth, QuikSCAT slice data are densely collected annually and posted at 1/120° (~1 km at the equator) in the latitude‐longitude geographic projection. Then, a mathematical transformation, the Rosette Transform, is applied to calculate an ensemble average of slice backscatter values, denoted as and expressed by

(2.3)

where N is the total number of slice data centered at the given location, and is the backscatter value measured at time t _i and azimuth direction φ _i of each slice. In (2.1), is the mean part obtained by the Rosette Transform, and ℜ is the residual from the zero‐mean fluctuating part of the radar backscatter. In an urban area with stationary buildings and structures that have strong radar return, has a large and stable value ℜ while becoming small with a sufficiently large number of samples collected over a year (Nghiem et al. 2009). DSM backscatter provides a method to observe urban spatial pattern and temporal change based on physical structure in 3D, such as houses, factories, industrial plants, commercial centers, freeways, bridges, etc. As such, it is a crucial input to physical‐based climate‐urban nested modeling to assess impacts of urbanization (Jacobson et al. 2015, 2019), in contrast to urban extent arbitrarily demarcated based on administrative, legislative, and/or political arrangements (Taubenböck et al. 2019).

Notably, recent works have validated the capability of QuikSCAT data processed using the DSM for built‐up volume estimation in urban areas (Nghiem et al. 2017; Mathews et al. 2019). As an example, high DSM backscatter values for the Los Angeles area (see Figure 2.8a) exhibit a similar spatial pattern with high amounts of urban built‐up volume (Figure 2.8b shown with individual building extents – i.e. clipped by building footprints). Similarly, the building volume and DSM patterns across large areas such as Austin, Texas (Figure 2.9) are nearly indistinguishable, demonstrating the capability of DSM to capture 3D building volume structures in the city (Nghiem et al. 2017).

FIGURE 2.8 Pattern of DSM backscatter for greater Los Angeles, California (a) with stronger backscatter response in areas with higher amounts of urban built‐up volume (red; see downtown) and a transition to orange/yellow for suburban and rural area. In the downtown area (b), greater building footprint volumes (white) are spatially collocated with higher backscatter values (dark orange, red).

To quantitatively evaluate the degree to which DSM data can account for urban built‐up volume, we compared DSM backscatter to lidar‐derived building volumes aggregated to the same scale (1 km grid). Specifically, QuikSCAT data were processed using the DSM to match the geographic extent of airborne lidar data for nine cities in the United States (see Table 2.1 for extents): Atlanta, Austin, Buffalo, Detroit, Los Angeles, New Orleans, San Antonio, Tulsa, and Washington, DC. The DSM data collapsed one year's worth of observations into a single dataset gridded in a pixel size of ~1 km. The year of data matched the year of lidar data collection (Table 2.1); the lidar data were captured on differing dates but all during leaf‐on conditions during the summer months.

FIGURE 2.9 Spatial trend patterns for Austin, Texas in 2006 overlaid on Google Earth basemap. (a) The pattern of lidar‐derived building volume with the rainbow color scale from 0 to 9 Mm³, and (b) the pattern of DSM radar backscatter with the rainbow color scale from −12 to −3 dB. The building volume and DSM patterns are nearly indistinguishable, demonstrating the capability of DSM to capture 3D building volume structures in the city (Nghiem et al. 2017).

Source: Nghiem et al. (2017).

Table 2.1 Coefficients of determination between DSM and lidar for nine US cities.

Source: Mathews et al. (2019)/Elsevier.

City	Analysis year	Analysis extent (km2)	DSM vs. lidar (r2)	DSM vs. lidar trend (r2)	DSM trend vs. lidar trend (r2)
Atlanta, GA	2003	79	0.13	0.33	0.77
Austin, TX	2006	390	0.21	0.72	0.98
Buffalo, NY	2004	342	0.14	0.38	0.69
Detroit, MI	2004	347	0.10	0.52	0.81
Los Angeles, CAa	2007	64	0.04*	0.26	0.64
New Orleans, LA	2008	346	0.04	0.21	0.33
San Antonio, TX	2003	640	0.20	0.75	0.97
Tulsa, OK	2008	1329	0.26	0.63	0.84
Washington, DC	2008	8297	0.32	0.66	0.98

Notes: r² is coefficient of determination in linear model. All correlations significant with p‐values < 0.01 unless otherwise noted (<0.05*).

^a Insufficient data sample due to limited extent of lidar data.

The 1 m spatial resolution lidar‐derived raster data were provided by the Army Geospatial Center. Datasets included a DTM and two DSMs (first‐return and last‐return versions) for all cities that were used to generate a last‐return DHM (calculated as DTM minus the lastreturn DSM) with relative heights from the ground surface. The last‐return dataset was selected to avoid noise introduced by vegetation in areas with lower building heights (e.g. suburban areas where trees overhang buildings). Data fusion incorporating building footprints to clip out building‐only areas (i.e. remove noise) within cities ensured volume values are representative of only built‐up volume. The building footprint approach though does not include infrastructure such as highway overpasses and similar large nonbuilding features. Further, building footprint data are not always readily available (i.e. from local government), and take significant time to generate from scratch by way of GIS hand digitizing or OBIA. All nonbuilding pixels were reassigned values of zero. Per‐pixel volume (m³) was then aggregated, summed, to match the 1 km spatial grid of the DSM data. The DSM and building volume values were then spatially coincident and prepared for statistical evaluation for measures of association, reported quantitatively as coefficients of determination or r². In addition, for comparison purposes, a spatial trend (see Şen 2017) of the data values for both the DSM and lidar was applied using the Trend tool in the ArcGIS Suite (e.g. DSM trend surface, lidar trend surface).

Results indicate a great deal of similarity between the DSM and lidar datasets. Figures 2.10 and 2.11 display two of the study cities, Tulsa and San Antonio, respectively, including all of the input datasets. The 1 km datasets illustrate the differences between the input data where the DSM (Figures 2.10 and 2.11c) is a smooth surface and the lidar version (Figures 2.10 and 2.11d) is much more discrete due to the extreme volume values representing the tallest buildings in the city's respective downtown and other highly built‐up areas. These differences provide visual confirmation of the differences between the two datasets and support the need for the spatial trend analysis. The spatial trend surfaces exhibit the same spatial patterns in terms of geographic extent and directionality of urban built‐up volume. The spatial trend concept is crucial as a quantitative representation of rural–urban transition that is a spatial change as a function of distance from the urban core (i.e. a spatial derivative). The spatial trend approach (Şen 2017) is mathematically robust, which circumvents the issue of taking spatial derivative of noisy discrete data of building structures, especially problematic and mathematically ill‐posed in very high‐resolution and thus very noisy data considering that buildings are typically vertical (e.g. 90° vertical walls) for which the derivative becomes infinitive or blows up. While many studies phrase the notion of rural–urban transition without a clear quantitative framework in a mathematically robust formulation, the DSM together with the spatial trend offers an elegant way to define and characterize rural–urban transition.

Statistical results provided in Table 2.1 indicate moderate, positive relationships between the DSM product and the aggregated lidar data following application of the spatial trend (i.e. DSM vs. lidar trend). Statistical results show weak correlations between raw comparisons (i.e. DSM vs. lidar) and strong, positive correlations between trend to trend comparisons (i.e. DSM trend vs. lidar trend). The highest correlation values (i.e. DSM trend vs. lidar trend) were observed for Austin, TX (r² = 0.98), Washington, DC (r² = 0.98), and San Antonio, TX (r² = 0.97; see Figure 2.11). Importantly, the direct, positive relationships are similarly predictive of built‐up volume in areas with low values (i.e. suburban, rural) as in areas with high values (i.e. downtown, industrial). As validated by building volume derived from lidar data with very limited availability in time and space, DSM can be applied for 3D monitoring of global urban areas (Nghiem et al. 2017; Mathews et al. 2019). Moreover, the linear relationship between DSM backscatter and 3D building volume holds true throughout the entire dynamic range of urban backscatter without a saturation effect (Mathews et al. 2019) either at the low limit (such as small wooden houses in residential areas) or the high end (such as steel skyscrapers in city centers). This implies that DSM can be used to estimate building volume density (i.e. the total build volume per pixel or per unit area) regardless of building size or type, or whether the total building volume consists of many small buildings or a few large buildings. Thereby, the DSM provides a simple approach applicable to various urban classes having different structural patterns without many confounding factors.

FIGURE 2.10 City and data extents along with raw and processed data and polynomial trend visualizations for Tulsa, Oklahoma for 2008: (a) reference map, (b) 1 m lidar first‐return DHM, (c) 1 km radar DSM, (d) 1 km aggregated lidar data (buildings only), (e) radar trend surface, and (f) lidar trend surface.

Importantly on a global scale, DSM has been used extensively in scientific research in both 2D and 3D for a diverse array of cities. For example, DSM analyses together with demographic data showed a positive relationship with ambient population distribution patterns in Bogotá in Colombia, Dhaka in Bangladesh, Guangzhou in China, and Quito in Ecuador (Nghiem et al. 2009). DSM urban observation products from satellites have been used to measure quantitatively the fourfold increase in urban extent of Beijing (Jacobson et al. 2015; Sorichetta et al. 2020), assess policy efficacy (i.e. clean air for the 2008 Olympics; Sorichetta et al. 2020), quantify changes in Beijing and Shanghai that have led to urban development policy (Nghiem 2015; Sorichetta et al. 2020), evaluate groundwater contamination in Italian cities (Masetti et al. 2015; Stevenazzi et al. 2015), estimate groundwater vulnerability in various future scenarios (Stevenazzi et al. 2017), and capture a real‐estate boom in Almaty (Groisman et al. 2017). Jacobson et al. (2019), using DSM urban data products as input to the Gas, Aerosol, Transport, Radiation, General Circulation, Mesoscale, and Ocean Model (GATOR‐GCMOM), found substantial impacts of urbanization on short‐term weather and pollution in two contrasting megacities of Los Angeles in the United States and New Delhi in India. In the Greater Saigon, Balk et al. (2019) paired census data with DSM and other built‐up layers finding that key indicators of economic development were more positively associated with volumetric changes in urban 3D patterns as detected by DSM than with 2D lateral expansion occurring on the urban periphery. DSM results revealed the formation of an extensive mega urban agglomeration along the Yangtze River and its vicinity consisting of Shanghai, Minhang, Baoshan, Jiading, Kunshan, Suzhou, Changshu, Wuxi, Nantong, Zhangjiagang, Jiangyin, Changzhou, Taizhou, Yangzhou, Nanjing, Ma'anshan, Wuhu, etc., along a swath extending over 450 km from the Pacific coast inland (Nghiem 2015). Over the Po Plain region in Italy, DSM was found to be more efficient than aerial surveys and census data for assessing groundwater vulnerability to pollution (Masetti et al. 2015). In sum, these works attest to the value of the DSM approach in application to urban analyses. The DSM is, therefore, capable of monitoring both small and large cities with differing building characteristics and built‐up patterns under diverse socio‐economic conditions as well as a variety of countries and continents under diverse geophysical conditions (e.g. deltas, valleys, mountains, tropics, deserts, coastal and inland regions, etc.) and climatic regions.

FIGURE 2.11 City and data extents along with raw and processed data and polynomial trend visualizations for San Antonio, Texas, for 2003: (a) reference map, (b) 1 m lidar last‐return DHM, (c) 1 km radar DSM, (d) 1 km aggregated lidar data (buildings only), (e) radar trend surface, and (f) lidar trend surface.