Estimation of Continuous Urban Sky View Factor from Landsat Data Using Shadow Detection

Sky View Factor (SVF, a dimensionless value between 0 and 1 representing obstructed and unobstructed sky, respectively) has an important influence on urban energy balance, and is a key contributor to the Urban Heat Island (UHI) effect experienced by heavily built up regions. Continuous urban SVF maps used in modeling the spatial distribution of UHI can be derived analytically using Lidar data; however, Lidar data are costly to obtain and often lack complete coverage of large cities or metropolitan areas. This study develops and validates a method for estimating continuous urban SVF from globally available Landsat TM data, based on the presence of shadows cast by SVF-reducing urban features. SVF and per-pixel shadow proportion (SP) were first calculated for synthetic grid cities to confirm a logarithmic relationship between the two properties; then Lidar data from four US cities were used to determine an empirical regression relating SP to SVF. Spectral Mixture Analysis was then used to estimate per-pixel SP in a Landsat 5 TM image covering the Greater Vancouver Area, Canada, and the empirical regression was used to calculate SVF from per-pixel SP. The accuracy of the resulting SVF map was validated using independent Lidar-derived SVF data (R2 = 0.78; RMSE = 0.056).


Introduction
The Urban Heat Island effect is a well-established phenomenon [1] with clear negative health impacts, especially during summer heat waves [2,3]. Sky View Factor (SVF) is defined as the ratio of radiation received from the sky by a planar surface to that received from the entire hemispheric radiating environment [4] and is typically simplified as a dimensionless value between 0 and 1, representing a completely obscured and unobscured sky, respectively [5]. SVF has been shown to have a significant effect on the Urban Heat Island ( [6][7][8][9][10][11], others), with obstructed areas delaying the escape of longwave radiation at night, resulting in slower cooling [12]. A daytime "cool island" effect can also occur due to the obstruction of incoming solar radiation, though its strength and duration are not as prevalent as the nocturnal effect [11]. A continuous map of SVF could therefore be a valuable tool in modeling urban energy balance, and by extension, be useful for mapping daytime and nighttime urban air temperatures [13].
Early methods for estimating SVF used the manual analysis of hemispherical photographs taken in situ using fisheye lenses [14]. Computer analysis improved the speed and accuracy of the process [8,15], and it is now a common way to gather SVF data in urban climate studies [10,16,17]. A more recently developed method involves the use of 3D modeling to determine a continuous Digital Surface Model (DSM) in ENVI, or downloaded from the source directly as a 5 m first-return raster DSM. ) was chosen to minimize differences in urban structure between the time of Landsat and validation data acquisition. An additional factor was the necessity of the image to have a moderate solar elevation (around 40 ¥ ) to optimize SP-SVF regression (discussed further in Section 3.1). The image was clipped to include only the study area Vancouver, shown in Figure 1. ETM+ imagery was not used due to the Scan Line Corrector malfunction. September 2011) was chosen to minimize differences in urban structure between the time of Landsat and validation data acquisition. An additional factor was the necessity of the image to have a moderate solar elevation (around 40°) to optimize SP-SVF regression (discussed further in Section 3.1). The image was clipped to include only the study area Vancouver, shown in Figure 1. ETM+ imagery was not used due to the Scan Line Corrector malfunction.

Validation Data
Lidar data for the city of Vancouver flown on 25 April 2013 were acquired as point-cloud data, and converted to 5 m first-return raster DSM in ENVI to match the calibration data.

Methods
Mapping of urban sky view factor was achieved in three main steps, outlined in Figure 2: (1) development of an empirical relationship between SP and SVF specific to the solar elevation of the Landsat data using the calibration data of the four US cities; (2) use of spectral mixture analysis to estimate per-pixel SP from the Landsat data; and (3) application of the empirical SP-SVF relationship to map SVF. The resulting SVF map was then validated using a SVF map derived from independent Lidar data.

. Validation Data
Lidar data for the city of Vancouver flown on 25 April 2013 were acquired as point-cloud data, and converted to 5 m first-return raster DSM in ENVI to match the calibration data.

Methods
Mapping of urban sky view factor was achieved in three main steps, outlined in Figure 2: (1) development of an empirical relationship between SP and SVF specific to the solar elevation of the Landsat data using the calibration data of the four US cities; (2) use of spectral mixture analysis to estimate per-pixel SP from the Landsat data; and (3) application of the empirical SP-SVF relationship to map SVF. The resulting SVF map was then validated using a SVF map derived from independent Lidar data. Overview of the modeling process to estimate sky view factor (SVF) from Landsat imagery. Shaded regions correspond to numbered steps described in Section 3.

Development of Theoretical and Empirical Relationships between SP and SVF
To assess whether SVF could theoretically be predicted from SP and, if so, what kind of relationship would exist between the two variables, a synthetic 3D grid city with 30 m × 30 m buildings and 30 m wide streets was developed, similar to that used by Yuan and Chen [30] in a SVF modeling study. The synthetic city was built by creating a raster in an image manipulation program where a value of 0 represented street level, and a value of 1 represented buildings. Raster values could then be multiplied by a range of "building heights" to generate varying urban topography, and thus varying SVF and SP. For this city, SVF was determined at 1 m resolution using a SVF modeling program [31] that calculates the vertical angle to the horizon at a user specified scanning radius in a specified number of directions; in this case a radius of 500 m for 104 directions was used. SVF was calculated by averaging over an area with equal amounts of building, street, and intersection segments to eliminate any effect from the spatial offset between a structure and its shadow, and hence between SVF and SP, caused by the fact that structures cast shadows in one direction but reduce the SVF around them equally in all directions. SP was determined by modeling angles using the hillshade tool in ArcGIS 10 at 1 m resolution to create a binary shadow image. SP was then calculated as the Overview of the modeling process to estimate sky view factor (SVF) from Landsat imagery. Shaded regions correspond to numbered steps described in Section 3.

Development of Theoretical and Empirical Relationships between SP and SVF
To assess whether SVF could theoretically be predicted from SP and, if so, what kind of relationship would exist between the two variables, a synthetic 3D grid city with 30 m ¢ 30 m buildings and 30 m wide streets was developed, similar to that used by Yuan and Chen [30] in a SVF modeling study. The synthetic city was built by creating a raster in an image manipulation program where a value of 0 represented street level, and a value of 1 represented buildings. Raster values could then be multiplied by a range of "building heights" to generate varying urban topography, and thus varying SVF and SP. For this city, SVF was determined at 1 m resolution using a SVF modeling program [31] that calculates the vertical angle to the horizon at a user specified scanning radius in a specified number of directions; in this case a radius of 500 m for 104 directions was used. SVF was calculated by averaging over an area with equal amounts of building, street, and intersection segments to eliminate any effect from the spatial offset between a structure and its shadow, and hence between SVF and SP, caused by the fact that structures cast shadows in one direction but reduce the SVF around them equally in all directions. SP was determined by modeling angles using the hillshade tool in ArcGIS 10 at 1 m resolution to create a binary shadow image. SP was then calculated as the proportion of 1 m shadowed elements to total 1 m elements within 30 m blocks of cells. SVF in the synthetic city was manipulated by varying the building heights, and the resulting SP was modeled for a number of solar elevations (20 ¥ , 40 ¥ , 60 ¥ , and 80 ¥ ). For simplicity, solar azimuth was kept constant at 135 ¥ , close to the typical solar azimuth for mid-latitude Landsat data in the northern hemisphere. This modeling showed that a theoretical relationship between SVF and SP does exist, and that it follows a logarithmic pattern.
Lidar data from the four US cities were used to create SP-SVF relationships for real cities with a variety of street patterns, building heights, open spaces, etc., based on a solar elevation of 40 ¥ and solar azimuth of 135 ¥ . A solar elevation of 40 ¥ was chosen to minimize shadow saturation while still allowing small objects to cast detectable shadow. SP was determined using the same method as for the synthetic city, but a different approach was needed for SVF due to spatial variation in a real city's SVF. In the northern mid-latitudes, structures cast shadows to the northwest during the mid-morning time of a Landsat overpass, but due to variations in building height as well as local topography, the spatial offset between SVF and SP is not easily quantified. In this study a 7 ¢ 7 low pass filter was therefore applied to both the SVF derived using the Zakšek model [27], and SP, allowing a comparison between SP and the average SVF in the filtered area. Other filter sizes were also examined (3,5,7,9,11), and a trade-off between SVF prediction accuracy and spatial resolution was found to exist, for which the 7 ¢ 7 filter was chosen as an acceptable compromise. For Landsat TM's 30 m spatial resolution, the 7 ¢ 7 filter extends between 105 m and 148 m from the central point, which is adequate to account for shadows cast by the majority of buildings in the cities used in this study when illuminated at a solar elevation of 40 ¥ , and still small enough to preserve some fine-scale spatial variations in urban SVF. The final SP-SVF relationship was determined by fitting a logarithmic curve to a combination of all four cities' data points. A separate relationship was modeled for use with the Landsat data using the specific illumination angles present at image acquisition (elevation = 43.93 ¥ , azimuth = 152.02 ¥ ). The resulting equation was used to calculate SVF from a SP map.

Estimation of SP and SVF
SP was determined using Spectral Mixture Analysis (SMA) of the Vancouver Landsat TM data. A Minimum Noise Fraction (MNF) transformation was first applied to the data to reduce sensor noise [32], with only the first three components being used in the analysis, the rest having been visually determined to contain mostly noise.
Partial unmixing methods were applied to the MNF components. Partial unmixing involves the use of only the target endmember, in this case shade, effectively reducing the mixture analysis to only two endmembers: "shadow" and "non-shadow", thus removing the need to account for every surface type in a complicated urban environment. The shade endmember was selected by finding the darkest pixel in the Near Infrared (Landsat TM band 4) in our study area (excluding large water bodies). Band 4 was used to avoid the selection of shaded vegetation, as vegetation reflects strongly in this band.
Large water bodies were masked out using an existing landcover classification map of the region. The classification map was created using the CART classification tree algorithm, with spatial covariates including spectral indices such as NDVI and NDWI derived from Landsat TM data, and morphometric indices derived from a DEM. The model was trained and validated from points generated by a detailed manual inspection of high resolution imagery.
A number of partial unmixing methods were empirically examined in the study area, and two were chosen for their predictive qualities. Matched Filtering (MF) [33] and Adaptive Coherence Estimator (ACE) algorithms were chosen, as implemented in ENVI. In the empirical examination of the two methods using a sample Landsat and DSM dataset in the study area, it was found that MF generally slightly overestimated and ACE generally slightly underestimated SP. The average results from these two methods was found to provide an adequately accurate SP prediction, and used in the remaining analysis.
Upon applying a 7 ¢ 7 Low Pass filter to the SP data to account for SP-SVF spatial offset, the data were converted to SVF using the empirical regression equation derived from the four US cities.

Validation
SP and SVF values used for validation were derived from the Vancouver Lidar data in the same way as for the four US cities, and smoothed using a 7 ¢ 7 low pass filter to match the smoothed Landsat data. A stratified random sample of 1000 validation pixels were chosen from the study area, 500 each from manually identified high SVF and a low SVF regions. This subsampling was done to generate a balanced validation data set, as the vast majority of the study area is comprised of suburban housing, which has a high SVF, with only the downtown core containing low-SVF commercial towers.

Development of an Empirical Relationship between SP and SVF
Theoretical modeling of SP and SVF in the synthetic city resulted in logarithmic relationships of the form SVF = a + b ¢ ln(SP ¡ c) and showed good fits that gradually decreased with increasing solar elevation ( Figure 3). This decrease was due to resolution problems with ArcGIS's hillshade tool causing multiple pairs of points to have identical SP values at different SVFs in the higher solar elevations. At high solar elevations, the change in SP for a particular change in SVF was very small compared to the same interval at low solar elevations. The hillshade tool was limited to a resolution of about 0.001 due to the method of resampling 1 m pixels into 30 m pixels, and thus very small changes in SP were not accurately quantified. At low solar elevations, a problem with shadow saturation was encountered, wherein a decrease in SVF would no longer correspond to an increase in SP because the area was already in full shade. This effect can be seen in Figure 3, where saturated points are represented by the shape outlines. The saturation point occurred at progressively lower SVFs as the solar elevation increased, however it always occurred at SP = 0.75, due to the averaging process and the city design, in which 75% of the city was "street level" and 25% was flat rooftops that are sunlit regardless of solar elevation. These results implied the existence of a range of moderate solar elevations for which the SP-SVF relationship could be best quantified; where solar elevation was high enough to prevent shadow saturation but low enough to ensure that SP changes for a given SVF change were large enough to be easily resolved. The ideal solar elevation needed to achieve this effect was determined to be approximately 40 ¥ . SP and SVF values used for validation were derived from the Vancouver Lidar data in the same way as for the four US cities, and smoothed using a 7 × 7 low pass filter to match the smoothed Landsat data. A stratified random sample of 1000 validation pixels were chosen from the study area, 500 each from manually identified high SVF and a low SVF regions. This subsampling was done to generate a balanced validation data set, as the vast majority of the study area is comprised of suburban housing, which has a high SVF, with only the downtown core containing low-SVF commercial towers.

Development of an Empirical Relationship between SP and SVF
Theoretical modeling of SP and SVF in the synthetic city resulted in logarithmic relationships of the form SVF = a + b × ln(SP − c) and showed good fits that gradually decreased with increasing solar elevation ( Figure 3). This decrease was due to resolution problems with ArcGIS's hillshade tool causing multiple pairs of points to have identical SP values at different SVFs in the higher solar elevations. At high solar elevations, the change in SP for a particular change in SVF was very small compared to the same interval at low solar elevations. The hillshade tool was limited to a resolution of about 0.001 due to the method of resampling 1 m pixels into 30 m pixels, and thus very small changes in SP were not accurately quantified. At low solar elevations, a problem with shadow saturation was encountered, wherein a decrease in SVF would no longer correspond to an increase in SP because the area was already in full shade. This effect can be seen in Figure 3, where saturated points are represented by the shape outlines. The saturation point occurred at progressively lower SVFs as the solar elevation increased, however it always occurred at SP = 0.75, due to the averaging process and the city design, in which 75% of the city was "street level" and 25% was flat rooftops that are sunlit regardless of solar elevation. These results implied the existence of a range of moderate solar elevations for which the SP-SVF relationship could be best quantified; where solar elevation was high enough to prevent shadow saturation but low enough to ensure that SP changes for a given SVF change were large enough to be easily resolved. The ideal solar elevation needed to achieve this effect was determined to be approximately 40°.  The city simulation process resulted in evidence that there existed a good logarithmic relationship between SVF and SP, and revealed problems that might also be encountered when developing the relationship in real cities, namely problems with resolution and shadow saturation.
The logarithmic relationship between SP and SVF seen for the synthetic city was also evident for the four US cities (Figure 4), and a logarithmic function fit to their combined data points achieved a good fit (R 2 = 0.84). Shadow saturation was not a problem here because most of the shadow-casting structures were too small to cause saturation at 40 ¥ solar elevation. Where structures were large enough (mostly in San Diego), the local variations in building height meant that the rooftops of nearby lower buildings would accommodate the extra shade cast by taller building. The city simulation process resulted in evidence that there existed a good logarithmic relationship between SVF and SP, and revealed problems that might also be encountered when developing the relationship in real cities, namely problems with resolution and shadow saturation.
The logarithmic relationship between SP and SVF seen for the synthetic city was also evident for the four US cities (Figure 4), and a logarithmic function fit to their combined data points achieved a good fit (R 2 = 0.84). Shadow saturation was not a problem here because most of the shadow-casting structures were too small to cause saturation at 40° solar elevation. Where structures were large enough (mostly in San Diego), the local variations in building height meant that the rooftops of nearby lower buildings would accommodate the extra shade cast by taller building. SVF in the four-cities model does not reach a maximum value of one, nor does it reach values close to zero as would be expected in a heavily built up area such as San Diego or downtown Indianapolis. SVF values in completely sunlit (SP = 0) pixels will not equal 1 because urban features will reduce SVF farther than their shadows are cast, thus a completely sunlit pixel may still experience a SVF reduction. Areas large enough for SVF to indeed equal 1 rarely occur in urban environments, meaning SP-SVF relationship at low SP values in Figure 4 can be considered accurate. The lack of very low SVF values is the result of a combination of the 30 m pixel size and spatial smoothing: most pixels in heavily built up areas will include elements from both streets (with low SVF values) and rooftops (with high SVF values).
The SP-SVF relationship modeled at the Landsat-specific illumination angle also resulted in good fit, with a slightly higher R 2 value than the four-cities model (R 2 = 0.85). The resulting equation, SVF = 0.330872 − 0.25827 × ln(SP + 0.13481), was used to calculate SVF in the Vancouver study area (see Section 4.3).

Estimation of SP
Selection of the darkest non-water pixel in Landsat band 4 resulted in a pixel in the downtown, low SVF region of Vancouver being used as the shadow endmember. Inspection using Lidar data from the area showed that the pixel was indeed located in a part of the city completely shaded by a large building, an ideal result.
Use of this endmember in both MF and ACE algorithms resulted in two SP images, with the MF image consistently underpredicting SP and ACE consistently overpredicting it. Averaging the results from the two individual algorithms led to a more accurate estimation of SP. When compared to SP validation data in Figure 5, the result shows good SP estimation, with an RMSE of 0.076 and an R 2 = 0.83. SVF in the four-cities model does not reach a maximum value of one, nor does it reach values close to zero as would be expected in a heavily built up area such as San Diego or downtown Indianapolis. SVF values in completely sunlit (SP = 0) pixels will not equal 1 because urban features will reduce SVF farther than their shadows are cast, thus a completely sunlit pixel may still experience a SVF reduction. Areas large enough for SVF to indeed equal 1 rarely occur in urban environments, meaning SP-SVF relationship at low SP values in Figure 4 can be considered accurate. The lack of very low SVF values is the result of a combination of the 30 m pixel size and spatial smoothing: most pixels in heavily built up areas will include elements from both streets (with low SVF values) and rooftops (with high SVF values).
The SP-SVF relationship modeled at the Landsat-specific illumination angle also resulted in good fit, with a slightly higher R 2 value than the four-cities model (R 2 = 0.85). The resulting equation, SVF = 0.330872 ¡ 0.25827 ¢ ln(SP + 0.13481), was used to calculate SVF in the Vancouver study area (see Section 4.3).

Estimation of SP
Selection of the darkest non-water pixel in Landsat band 4 resulted in a pixel in the downtown, low SVF region of Vancouver being used as the shadow endmember. Inspection using Lidar data from the area showed that the pixel was indeed located in a part of the city completely shaded by a large building, an ideal result.
Use of this endmember in both MF and ACE algorithms resulted in two SP images, with the MF image consistently underpredicting SP and ACE consistently overpredicting it. Averaging the results from the two individual algorithms led to a more accurate estimation of SP. When compared to SP validation data in Figure 5, the result shows good SP estimation, with an RMSE of 0.076 and an R 2 = 0.83. There is a very slight SP overprediction at low SP values, which should lead to a slight underprediction of SVF in high-SVF areas.

Sky View Factor Image and Validation
The calculation of SVF from the SP image resulted in an SVF prediction showing a good fit to validation data, with R 2 = 0.78 and an RMSE of 0.056 ( Figure 6). As expected from examination of the SP data above, the model appears to slightly underpredict SVF in high SVF areas while slightly overpredicting in low SVF areas.

Sky View Factor Image and Validation
The calculation of SVF from the SP image resulted in an SVF prediction showing a good fit to validation data, with R 2 = 0.78 and an RMSE of 0.056 ( Figure 6). As expected from examination of the SP data above, the model appears to slightly underpredict SVF in high SVF areas while slightly overpredicting in low SVF areas. There is a very slight SP overprediction at low SP values, which should lead to a slight underprediction of SVF in high-SVF areas. Figure 5. Accuracy of shadow proportion (SP) values predicted using an average of results from Matched Filtering and Adaptive Coherence Estimator SMA algorithms, as compared to SP values derived analytically from Lidar artificially illuminated at the Vancouver Landsat data's solar elevation and azimuth conditions. SP is well predicted, with an RMSE of 0.076 and an R 2 = 0.83, though there exists a slight overprediction at low SP values. Highly overpredicted outliers are pixels representing grass fields; the cause if this is discussed in Section 5.

Sky View Factor Image and Validation
The calculation of SVF from the SP image resulted in an SVF prediction showing a good fit to validation data, with R 2 = 0.78 and an RMSE of 0.056 ( Figure 6). As expected from examination of the SP data above, the model appears to slightly underpredict SVF in high SVF areas while slightly overpredicting in low SVF areas. Figure 6. Accuracy of the predicted sky view factor (SVF) values as compared to SVF derived analytically from Lidar data. SVF is well predicted, with an RMSE of 0.056 and an R 2 = 0.78, These data correspond to the SVF images shown in Figure 7, using the darkest pixel in NIR Landsat band 4 in the Spectral Mixture Analysis. Figure 6. Accuracy of the predicted sky view factor (SVF) values as compared to SVF derived analytically from Lidar data. SVF is well predicted, with an RMSE of 0.056 and an R 2 = 0.78, These data correspond to the SVF images shown in Figure 7, using the darkest pixel in NIR Landsat band 4 in the Spectral Mixture Analysis.
The Landsat-derived SVF map is shown in Figure 7 along with SVF derived from validation Lidar data. The model captures the broad spatial SVF pattern in the study area, with major zones of low and high SVF being readily identified. Downtown Vancouver (Annotation "2" in Figure 7) is correctly predicted to have a much lower SVF than the surrounding residential areas, while smaller variations in these residential areas are also evident. The generally lower SVF in the Eastern regions of Vancouver are correctly predicted, as well as areas of low SVF in the southeast caused by trees. The Landsat-derived SVF map is shown in Figure 7 along with SVF derived from validation Lidar data. The model captures the broad spatial SVF pattern in the study area, with major zones of low and high SVF being readily identified. Downtown Vancouver (Annotation "2" in Figure 7) is correctly predicted to have a much lower SVF than the surrounding residential areas, while smaller variations in these residential areas are also evident. The generally lower SVF in the Eastern regions of Vancouver are correctly predicted, as well as areas of low SVF in the southeast caused by trees.  The large, obvious low-SVF square in south-central Vancouver, as well as smaller scattered patches are possibly the result of a discrepancy in scale between the Landsat shadow detection and the Lidar SVF calculations, where very small shadows such as those cast by individual blades of grass are not represented in the 5 m cell size from which the Lidar SVF is calculated. This issue and a possible solution are discussed in greater detail below.

Model Limitations
The key limitation for this SVF mapping method is the difference between the very small scale on which shadow and SVF reduction occurs, and the relatively larger scale that is relevant for SVF application in energy balance models, an issue best illustrated by the prediction of very low SVF over grass fields that is evident in Figure 7. Many small shadows cast by shrubs or large blades of grass may cause a grassy area to contain a high proportion of shadow, which would produce a Landsat prediction of low SVF for that area. However, at the metre scale that is typical of urban energy balance models, a similar scale to that of typical Lidar data (including our validation data), such small shadows would remain undetected. This problem can be solved by identifying grassland areas using a landcover classification and either applying a correction factor or assigning them a predetermined value (as was done for the ocean), though such reclassification was not done here in the interest of clarity.
Another important limitation is the inability of the model to predict SVF over water. This is caused by the spectral similarity between water and shaded urban areas, leading water to be identified as shade and thus leading to incorrect predictions of extremely low SVF on water. This issue can be resolved by identifying large water bodies using a landcover classification and replacing SVF values over water with either a constant value, as done in our study, or with values calculated using a bare-earth DEM to account for the reduction of SVF near shore.
A third limitation for this approach to Landsat-based SVF mapping is the necessity of the modeled city to be on relatively flat ground. Extreme topography will generate its own shadows, and even moderate topography will alter the length of shadows for a given value of SVF, thus changing the established SP-SVF regression. As the majority of the world's large cities are not located in areas of dramatic terrain, this limitation does not greatly affect its general applicability, although it should be kept in mind if application for cities in mountainous regions is considered.

Applicability of the Method to Cities Worldwide
These methods have been developed using only cities with a grid layout where streets generally run North-South and East-West. Grid cities are typical of North America, but cities elsewhere may follow a much less regular pattern. The general approach outlined in this study is likely to also work in cities with less regular street patterns, although the modeling of a separate SP-SVF relationship using Lidar data from a group of such cities will likely be required.

Use of a Synthetic City for Calibration
Using a synthetic city to model the SP-SVF relationship instead of multiple sets of Lidar data was attempted as a part of this study, though the results were unsatisfactory. Many varieties of regular grid cities were examined for their representability of real-world cities, including cities with varying street-to-building width ratios and varying building sizes, but these all showed poor results by not closely corresponding to the empirical relationship from real cities shown in Figure 4. This suggests that the SP-SVF relationship is influenced by the complex structural heterogeneity of real cities, which includes a range of street orientations, widths and shapes as well as local variations in building heights. Future research may find that a particular synthetic city design is better suited for equation calibration, or even that it would be possible to select a particular city type, for example to differentiate between North American and European style cities.

Potential Application
With increasing availability of Lidar data for many cities, from which accurate urban structure metrics including SVF can be derived, the need for a Landsat-based SVF mapping algorithm may seem small. We suggest three scenarios in which our algorithm is likely to provide the best available solution to SVF mapping. (1) Lidar data are still not available for many cities worldwide, or may be available only at a cost that makes them effectively inaccessible for a range of users while Landsat data are freely available for download; (2) Where Lidar data are freely available, they are unlikely to provide complete coverage of an extensive metropolitan area. Where such an area is the entity under study, Lidar-and Landsat-derived SVF maps may be combined to ensure complete coverage; (3) Any analysis of long-term change in urban structure must rely on temporally consistent and spatially extensive data sets. The Landsat image archive, with acquisitions by TM/ETM+/OLI sensors going back to 1984, provides three decades of freely available and spatially and spectrally consistent data, from which analyses of urban structure changes may be performed using our approach. In these three scenarios, there is currently no alternative approach for a spatially complete derivation of SVF.
Since the SVF data are derived from 30 m Landsat pixels, they are of a coarse resolution, being able to identify large-scale variation across cities and large metropolitan areas, but are unable to resolve intra-and inter-street scale SVF variations. As a result, appropriate applications will typically involve its use in macro-scale urban studies that investigate climate variability across these large scales. An example of such an application can be seen in its use as a data layer in a statistical model to estimate the daily maximum air temperature [34] in the Greater Vancouver Area. By including the SVF data along with other predictors in this regression model, the mean absolute errors of predictions for validation locations were reduced from 1.89 ¥ C to 1.82 ¥ C, with relatively greater improvement in densely built up areas that were poorly predicted without the SVF layer. In addition, permutation-based variable importance measures [35][36][37], which quantify the increase in prediction errors when a predictor is permuted in the validation data, suggest that the contribution of the SVF data layer to predictions made by the model is greater than that of other more commonly used predictors, such as vegetation liquid water and elevation. This provides a tangible example of the importance of SVF for the urban radiation and energy balance. Hemispherical photography and airborne Lidar methods, though able to generate SVF data with greater accuracy and at higher spatial resolution, have not been able to generate complete and continuous coverage of SVF data for a large urban environment, as is necessary for most mapping exercises. The approach described in this paper, which is based on freely available Landsat data, is the first that enables generation of such continuous coverage.

Conclusions
Analytical methods for calculating continuous sky-view factor (SVF) maps using 3D structural data have become commonplace, however 3D data can be expensive or difficult to acquire, resulting in many urban areas requiring the labour-intensive gathering of field-based point data. This study developed a method for producing spatially continuous SVF estimates from Landsat imagery based on the principle that shadows are cast by the same buildings that obscure the sky from view and thus reduce SVF. We first confirmed the existence of a logarithmic relationship between shadow proportion (SP) and SVF through simulations using a synthetic grid city, which, when artificially illuminated, allowed us to investigate SP and SVF values as they vary with changing sun elevation and building heights, and then derived an empirical logarithmic relationship between SP and SVF for real cities, calibrated using Lidar data from four US cities. We then estimated SP using the averaged result from two partial unmixing approaches, Matched Filtering and Adaptive Coherence Estimation, applied to Landsat 5 TM data from Vancouver, Canada. Both approaches effectively simplify target endmembers to "shadow" and "non-shadow" and thus eliminate the need to define many endmembers for the complicated urban landcover in the area. The shadow endmember was automatically defined as the darkest non-water pixel in Near Infrared (Landsat band 4). The resulting SP estimates closely matched SP values derived from independent validation Lidar data covering areas with a range of urban structures (RMSE = 0.076, R 2 = 0.83). Finally, we applied the empirical SP-SVF relationship to the SP estimates to produce per-pixel estimates of SVF for Vancouver. These Landsat-based SVF estimates also closely matched SVF values derived from the Lidar data (RMSE = 0.056, R 2 = 0.78). We suggest that the methodology described and validated in our study may be the best approach to generate spatially complete information on sky-view factor in urban environments whenever Lidar data are unavailable or their coverage is spatially or temporally incomplete.
Supplementary Materials: The following are available online at www.mdpi.com/2072-4292/8/7/568/s1, Table S1: Equation coefficients for SP-SVF empirical regression equation of the form SVF = a + b ¢ ln(SP ¡ c) at various common solar positions. Solar azimuths between 135 and 160 are common in North American Landsat images, and solar elevations near 40 ¥ have been determined to be optimal for SP-SVF regression application. This data set was created by modelling the SP-SVF relationship in Lidar data from four North American cities (Listed in Table 1 of the original manuscript), and can be applied directly to a smoothed shadow proportion image.