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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