The GRAZ Method—Determination of Urban Surface Temperatures from Aerial Thermography Based on a Three-Dimensional Sampling Algorithm

Abstract

A novel method to derive surface temperatures from aerial thermography is proposed. Its theoretical foundation, details regarding the implementation, relevant sensitivities, and its application on a day and night survey are presented here. The method differs from existing approaches particularly in two aspects: first, a three-dimensional sampling approach is used to determine the reflected thermal radiation component. Different surface classes based on hyperspectral classification with specific properties regarding the reflection and emission of thermal radiation are considered in this sampling process. Second, the method relies on a detailed, altitude-dependent, directionally and spectrally resolved modelling of the atmospheric radiation transfer and considers the spectral sensitivity of the sensor used. In order to accurately consider atmospheric influences, the atmosphere is modelled as a function of altitude regarding temperature, pressure and greenhouse gas concentrations. The atmospheric profiles are generated specifically for the time of the survey based on measurements, meteorological forecasts and generic models. The method was initially developed for application in urban contexts, as it is able to capture the pronounced three-dimensional character of such environments. However, due to the detailed consideration of elevation and atmospheric conditions, the method is also valuable for the analysis of rural areas. The included case studies covering two thermographic surveys of city area of Graz during daytime and nighttime demonstrate the capabilities and feasibility of the method. In relation to the detected brightness temperatures apparent to the sensor, the determined surface temperatures vary considerably and generally cover an increased temperature range. The two processed surface temperature maps of the city area of Graz are finally used to validate the method based on available temperature recordings.

1. Introduction

Amid global warming and rising temperatures, especially in urban areas, aerial thermography is increasingly applied to identify, monitor or optimize local overheating phenomena related to urban- (UHI) or intraurban-heat-island (IUHI) effects. For this purpose, the acquired thermographic images taken by sensors on planes or satellites are still predominantly used and interpreted directly. This means that the radiance measured by each pixel of the sensor is converted into brightness temperature by inverting Planck’s law for blackbody radiance. However, the temperatures obtained using this approach usually differ significantly from the thermodynamic temperatures of the mapped surfaces [1]. This can mainly be attributed to two reasons. First, the atmospheric path of the thermal radiation has to be considered. Thermal radiation emitted from the surface is attenuated by absorption and scattering of atmospheric gas molecules. Additionally, the thermal radiation emitted by the atmospheric gases is detected by the sensors (upwelling radiation). Second, the observed surfaces will generally not exhibit a blackbody emission profile. In order to determine the thermodynamic temperature of a non-blackbody, the surface needs to be characterized by its emissivity and further properties, and based on these parameters, the radiation incident to the surface and reflected to the sensor has to be determined.

Atmospheric corrections are widely applied and based on a variety of models and data sources. Most approaches were developed for the field of spaceborne remote sensing (see, e.g., [2]). Consequently, additional spectral information gathered by satellites is often used in order to establish atmospheric radiative transfer models for the spectral range of thermal radiation. The models are primarily used to determine land surface temperatures (LST) based on land surface emissivities (LSE) and usually consider only vertical atmospheric paths. LST and LSE are commonly defined as average values characterizing the distribution of surface properties mapped onto a single pixel of the detector [3]. While LST models are useful for climate research and meteorology, their generally coarser spatial resolution limits their value for performing research focused on urban climate. However, currently available modern surveying methods based on photogrammetry or Lidar can provide city models with a high level of detail. Combined with surface classification, usually based on hyperspectral information, the high spatial resolution allows the creation of city models that consist of distinctively oriented subsurfaces of homogeneous or quasi-homogeneous material classes, e.g., pavements, trees, grass, specific roof types etc. In order to exploit this comprehensive spatial information, it is necessary to perform thermal analysis on the level of single surface temperatures. The more detailed model allows for a more sophisticated determination of surface temperatures by considering the local environment, surface orientations and classifications. In contrast, most LST approaches imply that the provided temperatures represent averages over many subsurfaces with different surface properties and/or orientations.

In recent years, several methods were proposed for evaluating spatially highly resolved thermal information of urban environments; see, e.g., [4,5,6]. These methods usually focus on singular applications and research questions; however, they commonly fall short of addressing all significant aspects of the complex radiation transfer processes. This is, however, required to evaluate or interpret thermographic information at the urban scale correctly. An exception to this is recent work (see, e.g., [7,8]) where the DART model is used to consider three-dimensional geometries and atmospheric influences. In these approaches, mostly forward raytracing of scattered thermal radiation and voxel-based model of the atmosphere is applied. Consequently, the term ‘multi-scattering’ is commonly used to refer to this approach. In the method proposed here, we utilize the principle of reversibility of the light (i.e., thermal radiation) and apply backward-raytracing originating from the sensor instead. Viewed from this perspective, we use the terms ‘reflected thermal radiation’ and ‘atmospheric influences’ and avoid the ambiguous term ‘multi-scattering’.

An appropriate method for the evaluation of thermographic data at an urban scale requires an approach that combines methods of remote sensing, in particular atmospheric corrections, with GIS data formats and methods, as well as with methods that can address the problem’s three-dimensional nature. The necessity for elaborate spatial processing arises from the fact that even if an ideal correction of the atmospheric influences is performed, the detected brightness temperatures of urban surfaces exhibit a significant deviation from thermodynamic temperatures. Compared to the natural terrain, urban environments are characterized by a more distinctive spatial structure and comprise artificial surfaces with low emissivity values (e.g., glass and metal roofs). Consequently, the detected thermal radiation over the urban environment contains significant amounts of reflected radiation that need to be determined. Hence, complex processing that utilizes additional data sources is required to derive urban surface temperatures with high accuracy.

We propose a new method that is, in particular, dedicated to determining surface temperatures at the urban scale. Thermographic information is processed utilizing digital surface models, ground classification, as well as the local atmospheric condition. The method is based on backward raytracing and specifically considers the three-dimensional nature of the relevant radiative exchanges. It comprises an atmospheric model that considers the varying elevation of the terrain, as well as the significant angular gradient of the downwelling radiation. It uses local meteorological data from weather stations, radiosonde measurements and local forecast data. We refer to the method as the “Graz method”, as all authors are located in the city, the first application was carried out here and the city supported our research.

2. Method

2.1. Method Overview

In order to capture the heterogeneity of the urban environment, we are applying a pixel-based local sampling process to determine (estimate) the relevant components to find the true surface temperature from thermography data by inversion of the radiation transfer equation. The process can be divided into four tasks. A schematic overview of the main components considered in the method which is provided in Figure 1. The components are referenced in brackets in the description of the four tasks below:

a. 

atmospheric modelling

Atmospheric modelling is required in order to determine the transmittance of the atmosphere (T), the upwelling radiation (U) and the downwelling radiation (D). Atmospheric profiles based on radiosonde data, local ground measurements as well as generic profiles are used to model the relevant atmospheric gas concentrations, as well as the temperature and pressure profile. Two important features of the method are as follows: firstly, for all three components, the local elevation (H) of each pixel processed, and associated with this, the distance to the sensor, is specifically considered. Secondly, the angular dependence of the downwelling radiation (D) is also determined and considered in the evaluation.

b. 

determination of surface properties and orientation

In order to evaluate the emitted and reflected thermal radiation, it is necessary to determine the surface properties (O). For this purpose, classification data based on hyperspectral data are used to assign a surface type to each pixel. Each surface type is defined by its emissivity value as well as an additional surface roughness parameter. The emissivity values of the surfaces are determined considering the sensor’s spectral sensitivity curve (S). The roughness parameter ranges from 0 to 1 and is used to take the reflective properties of the surface into account. A roughness parameter of 1 indicates a perfectly diffuse reflecting surface, whereas a surface with a value of 0 is modelled as a specular (mirror-like) reflector (e.g., flat glass or polished metals). For both models, the local orientation of the surface is considered. It is determined based on the area’s digital surface model (DSM).

c. 

determination of the incident radiation

As the sensor does not only detect thermal radiation emitted by the surface but also thermal radiation reflected by it, it is necessary to determine the relevant incident radiation. This is done by performing a sampling algorithm that considers the (nadir) viewing angle, the local surface orientation and the roughness of the surface. In the process, the viewing factors of the reflecting classes are determined. The following classes are considered: urban surface (X), vegetation surface (V), remote environment (R) and ten different angular segments of the sky (D). This refinement allows a more accurate estimation of the reflected thermal radiation component than most standard approaches. It is based on likely estimates for the outgoing thermal radiation of 13 different classes reflected by the evaluation point. Firstly, it is based on the assumption that any urban surface in proximity is likely to have a temperature close to that of the evaluation point. Secondly, the vegetational surfaces are assumed to exhibit a temperature close to the air temperature (see [9] and Section 2.8.3). Any remote environment is also considered to exhibit air temperature due to the attenuation of the thermal radiation in the atmosphere and its potentially high proportion of vegetation cover. Finally, ten different sky temperatures are applied due to their pronounced angular gradient. These temperatures can be determined for clear skies using atmospheric radiation transfer models (see Section 2.2).

d. 

solving the radiation transfer equation

Once all relevant components are evaluated for each pixel, its surface temperature can be determined by solving the radiation transfer equation that is derived in Section 2.3.

Key features of the proposed method are as follows:

• Elevation dependency of atmospheric parameters (transmittance, upwelling and downwelling radiation);
• Angular dependency of incident atmospheric radiation (“sky temperature gradient”);
• Orientation-dependent sampling of incident radiation;
• Consideration of different surface classes for sampling the incident radiation;
• Consideration of local emissivity and surface roughness;
• Consideration of spectral sensor sensitivity for atmospheric transmission, emission and surface emissivity;
• Consideration of local atmospheric profiles;
• Consideration of remote environment (distant terrain).

2.2. Atmospheric Model—Fundamentals and Relevant Dependencies

The spectrally resolved absorption and emission of the atmospheric gases in the relevant spectral range of the detector have to be determined. As commonly applied, the concentrations of five gases are considered for this purpose. Due to their interaction with thermal radiation, the gases are also referred to as greenhouse gases. The molecules considered are water vapor (H₂O), carbon dioxide (CO₂), methane (CH₄), nitrous oxide (N₂O), and ozone (O₃). The concentration of each gas component is modelled as a function of altitude. Beyond that, altitude-dependent profiles of the air pressure and air temperatures are required. Meteorological data are used to model the water vapor concentration and temperature profiles. In order to achieve the highest potential accuracy, data from different sources are merged. Data from ground measurement stations at different altitudes are used to model the profiles at low altitudes. Radiosonde measurements and/or local forecast data are evaluated to model mid-altitude profiles (to approx. 20.000 m altitude). Finally, generic profiles are used to model the high-altitude regions (up to 50.000 m). The three-component approach reflects the decreasing dynamics in the atmosphere. Hourly or even sub-hourly measurement data are required to model the dynamic heating or cooling at low, near-ground altitudes. Radiosonde data that are usually acquired daily or bi-daily can be used to model the variations at mid-altitudes, as surveys are usually carried out during relatively stationary weather conditions. Finally, constant profiles can be used to model the stratospheric layers beyond 20.000 m. It should also be noted that the relevance of the layers and, therefore, the requirements for precise modelling decrease with altitude because the absolute gas concentrations decrease along with the atmospheric pressure. The atmospheric pressure profile can be modelled accurately by fitting the parameters of the barometric pressure formula based on the radiosonde measurements.

The concentration of the four other gas components is mostly modelled based on seasonal generic profiles as they generally show less variation. However, if available, the concentration in the lowest atmospheric layer can be scaled based on ground measurements.

Once the concentration profiles of the gas components as well as the temperature and pressure profile are established, they can be used to model the interaction of the atmosphere with thermal radiation. In the spectral range of thermal radiation, the absorption and emission of atmospheric gases are mostly linked to rotational and vibrational transitions of the molecules, see, e.g., [10]. The corresponding radiation transfer equations can be solved with suitable tools, such as MODTRAN [11,12]. The atmospheric modelling tools allow for the determination of the emission and transmission by integration along predefined atmospheric paths in a spectrally resolved way. For the present application, the following components are of interest (see Figure 2):

Upwelling radiation (U): the upward directed atmospheric radiation along a vertical path reaching the sensor, from ground level to flight altitude.

Transmittance (T): the atmospheric transmittance for the thermal radiation originating from the ground and detected at the airborne sensor.

Downwelling radiation (D1–D10): the downward directed radiation from an infinite distance (space) to ground level at various angles.

All components are integrated considering the sensor’s spectral sensitivity (S).

While commonly only one downwelling component (hemispherical or normal) is used, this method proposes considering downwelling radiation originating from N different directions, as the corresponding radiance values vary with the viewing angle. The directions used to evaluate the downwelling radiance are not distributed uniformly on the angular scale. Instead, the corresponding cosines are distributed with equal spacing (see Equation (1) and Figure 2)

$${\theta }_i=arccos\left(\left(i-0.5\right)/N\right)\hspace{1em} {\theta }_{i,min}=arccos\left(\left(i-1\right)/N\right)\hspace{1em} {\theta }_{i,max}=arccos\left(i/N\right)$$

(1)

Following this definition, the resulting angular segments covered by one direction become significantly smaller towards the horizon. If N = 10 angles are considered, the lowest direction (D1) covers an angular segment of 4.3°, while the topmost direction (D10) ranges over 25.8°. This definition leads to several advantages. Firstly, as the length of the path through the atmosphere increases approximately proportional to $1/\cos \theta$, the gradient of the downwelling radiance increases significantly towards the horizon. Hence, it is useful to distribute the sampling directions more densely in this region. Secondly, the definition of the angles implies that the solid angles of each angular segment covered by each direction are equal. Finally, and most importantly, no trigonometric functions are required to assign a sky segment $i$ to a given sampling (viewing) direction $d$ that is provided in vector form (see below). The corresponding segment can simply be determined as follows:

$$i=\mathit{min}\left(1+\left[d.z\ast N\right],N\right)$$

(2)

The square brackets in Equation (2) indicate an integer rounding operation. The equation can be evaluated with high computational performance, which is essential for the sampling process that is applied in this method.

All atmospheric parameters are determined in a spectrally resolved way and then integrated using the sensor’s spectral sensitivity curve $S$.

While the radiance for the upwelling radiation can be integrated directly by applying Equation (2), the spectrally resolved transmittance has to be considered as well in the integration (see in Section 2.3).

$$L=\int S\left(\lambda \right) L_{\lambda }\left(\lambda \right) d\lambda$$

(3)

The spectrally resolved radiance values for the upwelling and downwelling radiation at different incidence angles are illustrated in Figure 3. Additionally, the spectral sensitivity of the sensor used in the application presented here is depicted. As can be seen, the downwelling radiation varies significantly with the angle of incidence. While the atmospheric window in the range of 8 µm to 14 µm is pronounced at the lowest angles near the zenith (LD10), it is increasingly saturated for higher angles close to the horizon (up to LD1). The upwelling spectral radiance resembles the downwelling component for low angles (LD1). The upwelling radiance values are, however, generally lower due to the shorter atmospheric path. Further, the significant peak of the ozone layer (around 9.5 µm) is missing, as it is located well above the flight altitude and, therefore, not part of the integration path for the upwelling radiation.

A key feature of the proposed method is the consideration of terrain elevation in atmospheric modelling. Especially for thermographic mapping of hilly or mountainous regions, it is crucial to consider the changing local elevation. In the application presented in this publication, the survey flight was performed at an altitude of 1675 m MSL (true altitude above mean sea level), while the elevation of the mapped urban region ranges from 330 m to 754 m MSL. Hence, the atmospheric path lengths are reduced by roughly a third relative to the longest atmospheric path, i.e., the lowest point. Beyond the path length variation, the air temperature variation with altitude further affects the radiance values for the up- and downwelling radiation. In order to capture the elevation dependence, the atmospheric path calculations, as described above, are carried out for a number of different elevations. The resulting values for the up- and downwelling radiation, as well as for the transmittance, then serve as interpolation points to derive functional representations of the corresponding quantities. We have determined that fourth-order polynomial fits are suitable for this purpose.

Figure 4 depicts the derived elevation dependence of the radiance after applying the sensor-related spectral integration based on Equation (3). As can be seen, the up- and downwelling radiances decrease with altitude due to the shorter atmospheric paths and the decrease in temperature with altitude. The near horizontal downwelling path (LD1) shows the highest total value, but also the highest gradient as the integrated gas concentration along its path is the highest.

The upwelling atmospheric radiance (LU) shows the highest relative decrease, as in the case of high elevations, and only a small atmospheric column remains between the sensor and the terrain. Figure 5 shows the elevation dependence of the transmittance parameter for blackbody radiation of different temperatures. It can be seen that the transmittance of the atmosphere is generally higher for higher brightness temperatures. The relative increase is approximately 1.9% per 100 m for the lower altitudes and it accelerates towards the sensor position.

Figure 4 and Figure 5 demonstrate that the elevation dependencies of the relevant radiances and transmittances are significant and should be considered for any survey performed over non-flat terrains. In the elevation range of the presented application (330–754 m MSL), the radiance values decrease by approximately 13%, while the transmittance value increases by about 8%. The actual values and gradients depend on the atmospheric profiles at the time of the survey, of course.

2.3. Determination of True Surface Temperature

The equation used to determine the surface temperatures can best be derived when the at-sensor spectral radiance is formulated; see Equation (4). For the sake of clarity, the $\left(x,y\right)$ argument is omitted on variables that depend on coordinates. Instead, a hat accent is added to these variables to indicate that they depend on discrete two-dimensional coordinates, i.e., that they represent a two-dimensional array. The at-sensor radiance comprises three different components: the thermal emission of the surface itself, the thermal radiation reflected on the surface and the upwelling radiation. The first two components are attenuated on their path to the sensor, while the upwelling radiance is, by definition, calculated for the sensor position. Therefore, the at-sensor radiance can be written as follows:

$$\begin{gathered} {\widehat{L}}_{\lambda ,sensor}\left(\lambda ,{\widehat{T}}_x,\widehat{h},{\widehat{\alpha }}_x,{\widehat{\alpha }}_y\right)={\tau }_{\lambda }\left(\lambda ,\widehat{h}\right)\left[{\widehat{L}}_{\lambda ,x}\left(\lambda ,{\widehat{T}}_x\right)+\left(1-{\widehat{\epsilon }}_{\lambda }\left(\lambda \right)\right){·\widehat{L}}_{\lambda ,inc}\left(\lambda ,{\widehat{\alpha }}_x,{\widehat{\alpha }}_y\right)\right]+L_{\lambda ,up}(\lambda ,\widehat{h}) \\ {\widehat{L}}_{\lambda ,x}\left(\lambda ,{\widehat{T}}_x\right)={\widehat{\epsilon }}_{\lambda }\left(\lambda \right)·B_{\lambda }\left(\lambda ,{\widehat{T}}_x\right) \end{gathered}$$

(4)

${\widehat{L}}_{\lambda ,sensor}\left(\lambda ,{\widehat{T}}_x,\widehat{h},{\widehat{\alpha }}_x,{\widehat{\alpha }}_y\right)$ … spectral radiance at sensor pixel [W/(cm2.sr.µm)]

${\widehat{L}}_{\lambda ,x}\left(\lambda ,{\widehat{T}}_x\right)$ … spectral radiance of surface [W/(cm2.sr.µm)] (to be determined)

${\tau }_{\lambda }\left(\lambda ,\widehat{h}\right)$ … spectral transmittance for point with elevation$\widehat{h}$ [m]

${\widehat{\epsilon }}_{\lambda }\left(\lambda \right)$ … spectral emissivity of surface at mapped point

$B_{\lambda }\left(\lambda ,{\widehat{T}}_x\right)$ … Blackbody spectral radiance of mapped point with temperature${\widehat{T}}_x$ [K]

${\widehat{L}}_{\lambda ,inc}\left(\lambda ,{\widehat{\alpha }}_x,{\widehat{\alpha }}_y\right)$ … incident spectral radiance [W/(cm2.sr.µm)] at mapped point. Function of terrain slope angles ${\widehat{\alpha }}_x$ and ${\widehat{\alpha }}_y$

$L_{\lambda ,up}(\lambda ,\widehat{h})$ … upwelling radiance for mapped point with elevation $\widehat{h}$ [m]

The term ${\widehat{L}}_{\lambda ,inc}$ comprises all components incident to the surface and further reflected to the sensor. Much effort is put into accurately modelling this incident radiance component in the proposed approach. For this purpose, we distinguish 13 different classes of incident radiation:

• Urban (non-vegetation) surface;
• Vegetation surface;
• Remote environment;
• 10 different segments of the sky.

Furthermore, three assumptions regarding the temperature of these surfaces are required. In the following, the term “locally visible” is used to refer to surfaces relevant to the incident radiation, as an observer on the determination point would be able to view these surfaces.

I. 

The temperature of locally visible urban, non-vegetation surfaces is identical to the temperature at the evaluation point.

While this assumption cannot be true in general, it represents a plausible estimate. The near local environment is generally the dominant source for the reflected thermal radiation, as it has the highest view factors. On the other hand, the proximity and high view factor imply a significant thermal exchange with the evaluation point that limits potential temperature differences (e.g., a street canyon with the surrounding facades).

II. 

The temperature of the locally visible vegetation surfaces is identical to the local air temperature.

Vegetation (i.e., trees, shrubbery and grass) generally exhibits very large surface-to-volume ratios that can be linked to a high heat transfer coefficient. Therefore, the surface temperature of vegetation cannot deviate significantly from the air temperature, as this would require (or produce) a vast amount of energy.

III. 

The temperature of the locally visible remote environment is identical to the local air temperature.

On the one hand, for most locations in moderate climates, it can be assumed that the considered ‘remote environment’ (i.e., mountains and hills) is likely covered with large proportions of vegetational surfaces and, therefore, the statements just made in II. apply. On the other hand, as discussed in Section 2.2, the atmosphere attenuates thermal radiation propagating in near horizontal directions. Hence, its brightness temperature converges towards the local air temperature.

A sampling method is used to determine the contribution of each of the 13 classes to the total incident radiance, $L_{\lambda ,inc}$. Local viewing-factors $\widehat{\omega }$ are computed for each evaluation point to weigh the individual components (see Section 2.6). In the currently implemented approach, the same temperature is assumed for the remote environment and the vegetation surfaces. While for geometrical reasons, it can be assumed that the remote environment, mostly hills or mountains, will generally be located at higher altitudes and, therefore, should exhibit lower temperatures, it has to be considered that the thermal radiation of the extended, almost horizontal atmospheric paths will shift the radiative temperature towards the local air temperature. Therefore, assuming the same local air temperature for the vegetation surfaces is a viable approximation. In general, the approach is based on a number of plausible assumptions that might not be valid for specific configurations. However, these assumptions are approximations that can generally increase the accuracy of the calculation. Even if these assumptions might seem crude at first glance, it has to be considered that the currently applied methods either assume a flat environment or, like most sky-view-factor (SVF)-based methods, implicitly assume the entire environment to exhibit the same temperature as the measurement point. Hence, the detailed sampling performed for every grid point to determine the local environment will generally lead to more accurate results. In particular, the implicitly included determination of the shape of the skyline and the view factors of the different sky segments are essential, as the “sky temperatures” (i.e., downwelling radiation) differ significantly from any ground-based thermal emissions.

$$\begin{gathered} {\widehat{L}}_{\lambda ,inc}\left(\lambda ,{\widehat{\alpha }}_x,{\widehat{\alpha }}_y\right)=\widehat{d}·{\widehat{\omega }}_{V/RE}·B_{\lambda }\left(\lambda ,T_{V/RE}\right)+\widehat{d}·{\widehat{\omega }}_x·B_{\lambda }\left(\lambda ,{\widehat{T}}_x\right)+\widehat{d}·{\sum }_{i=1}^{10}{\widehat{\omega }}_{sky,i}{L}_{\lambda ,sky}\left(\lambda ,{\theta }_i\right) \\ +(1-\widehat{d})·L_{\lambda ,spec}(\lambda ,\widehat{h},\widehat{j}({\widehat{\alpha }}_x,{\widehat{\alpha }}_y\left)\right) \end{gathered}$$

(5)

$\widehat{d}$ … diffuseness of surface [0..1]; 0: ideal specular reflection/1: ideal diffuse reflection (Lambert)

${\widehat{\omega }}_{V/RE}$ … View-factor for vegetation surfaces and remote environment

${\widehat{\omega }}_x$ … View-factor for urban surfaces

${\widehat{\omega }}_{sky,i}$ … View-factor for ith sky segment (see Figure 2 and Equation (1))

$T_{V/RE}$… Temperature assumed for vegetation and remote environment

${\widehat{T}}_x$… Local urban surface temperature (to be determined)

$L_{\lambda ,sky}\left(\lambda ,{\theta }_i\right)$… spectral radiance of sky segment i, i.e., downwelling radiation, see Section 2.2

${\widehat{L}}_{\lambda ,spec}(\lambda ,\widehat{j})$… spectral radiance of specularly reflected incident radiation from surface class$\widehat{j}$ , see Section 2.2

${\widehat{\alpha }}_x,{\widehat{\alpha }}_y$ … slope angles of the surface at the evaluation point

Note that the radiance for the vegetation and remote environment surfaces as well as for the urban surfaces, are not scaled by emissivity factors, i.e., blackbody emitters are assumed. For non-black-body emitters ($\epsilon <1$), it is necessary to determine the reflected thermal radiation in order to determine the entire radiation originating from the surface. The currently implemented approach assumes that the reflected fractions saturate the emitted thermal radiation to achieve a blackbody spectrum.

In order to determine the specularly (i.e., mirror-like) reflected component $L_{\lambda ,spec}$, the class that is mirrored to the sensor by specular reflection is determined for every evaluation point (see Section 2.6). A class index $j$ of −1 indicates that the classes ‘remote environment’ or ‘vegetation’ are reflected, 0 indicates that local urban environment is reflected, while a value in the range of 1 to 10 refers to a specific sky segment. Consequently, the specularly reflected spectral radiance is defined as either the blackbody radiance of the relevant temperature or as the downwelling radiance of the relevant sky segment:

$${\widehat{L}}_{\lambda ,spec}\left(\lambda ,\widehat{h},\widehat{j}\right)=\left\{\begin{array}{c}j=-1:\hspace{1em}{B}_{\lambda }\left(\lambda ,T_{V/RE}\right)\\ j=0:\hspace{1em}{B}_{\lambda }\left(\lambda ,{\widehat{T}}_x\right)\\ j=1..10:\hspace{1em}{L}_{\lambda ,sky}\left(\lambda ,{\theta }_i\right)\end{array}\right.$$

(6)

The spectral radiance equations provided so far could be used to determine the surface emission in a spectrally resolved way, potentially achieving the highest accuracy. Most thermal sensors, however, provide only integrated radiance values for a specific wavelength range, like the commonly used microbolometers. Hence, it is necessary to introduce the spectrally integrated components into the provided equations. In order to do this, two different types of integration have to be applied. While the upwelling radiance and sensor radiance can be integrated using solely the sensor’s spectral sensitivity function, the spectral transmittance has to be considered for all other components as their radiance is attenuated on the atmospheric path towards the sensor. In order to indicate this, a τ superscript is added for these components. The spectral integration is based on the grey-body assumption for the temperature of the surface that is being determined. This implies that an averaged scalar value for the emissivity has to be used instead of the spectral emissivity function. However, the sensor’s sensitivity range and the average expected temperature should be considered when defining the emissivity. The spectrally integrated version of Equation (4) with Equation (5) substituted can be written as follows:

$${\widehat{L}}_{sensor}\left({\widehat{T}}_{sensor}\right)=\widehat{\epsilon } {\widehat{L}}_{\tau ,x} \left({\widehat{T}}_x,\widehat{h}\right) + \left(1-\widehat{\epsilon }\right)\left[\begin{array}{c}\widehat{d}·{\widehat{\omega }}_{V/RE}·L_{\tau ,V/RE}\left(\widehat{h},T_{V/RE}\right)\\ \hspace{1em}+\widehat{d}·{\widehat{\omega }}_x·{\widehat{L}}_{\tau ,x} \left({\widehat{T}}_x,\widehat{h}\right)\\ +\widehat{d}·{\sum }_{i=1}^{10}{\widehat{\omega }}_{sky,i}{L}_{\tau ,sky,i}\left(\widehat{h}\right) \\ \hspace{1em}\hspace{1em}\hspace{1em} +\left(1-\widehat{d}\right)·L_{\tau ,spec}(\widehat{h},\widehat{j})\end{array}\right]+L_{up}\left(\widehat{h}\right)$$

(7)

with the spectrally integrated components:

$${\widehat{L}}_{sensor}\left({\widehat{T}}_{sensor}\right)={\int }_{\lambda }S\left(\lambda \right) B_{\lambda }\left(\lambda ,{\widehat{T}}_{sensor}\right) d\lambda$$

(8)

$${\widehat{L}}_{up}\left(\widehat{h}\right)={\int }_{\lambda }S\left(\lambda \right) L_{\lambda ,up}\left(\lambda ,\widehat{h}\right) d\lambda$$

(9)

$${\widehat{L}}_{\tau ,x} \left({\widehat{T}}_x,\widehat{h}\right)={\int }_{\lambda }S\left(\lambda \right) \tau \left(\lambda ,\widehat{h}\right) B_{\lambda }\left(\lambda ,{\widehat{T}}_x\right) d\lambda$$

(10)

$${\widehat{L}}_{\tau ,V/RE}\left(\widehat{h},T_{V/RE}\right)={\int }_{\lambda } S\left(\lambda \right) \tau \left(\lambda ,\widehat{h}\right) B_{\lambda }\left(\lambda ,T_{V/RE}\right) d\lambda$$

(11)

$${\widehat{L}}_{\tau ,sky,i }\left(\widehat{h}\right)={\int }_{\lambda } S\left(\lambda \right) \tau \left(\lambda ,\widehat{h}\right) L_{\lambda ,sky}\left(\lambda ,{\theta }_i\right) d\lambda$$

(12)

$${\widehat{L}}_{\tau ,spec}\left(\widehat{h},\widehat{j}\right)=\hspace{1em}\left\{\begin{array}{c} j=-1:\hspace{1em}{\widehat{L}}_{\tau ,V/RE}\left(\widehat{h},T_{V/RE}\right)\hfill \\ j=0:\hspace{1em}{\widehat{L}}_{\tau ,x} \left({\widehat{T}}_x,\widehat{h}\right)\hfill \\ j=1..10:\hspace{1em}{\widehat{L}}_{\tau ,sky,j }\left(\widehat{h}\right)\hfill \end{array}\right.$$

(13)

In order to determine the desired quantity of the surface temperature ${\widehat{T}}_x$, Equation (7) is solved for ${\widehat{L}}_x$. For the common case, the solution is as follows:

$${\widehat{L}}_{\tau ,x}\left({\widehat{T}}_x,\widehat{h}\right)={\displaystyle \frac{{\widehat{L}}_{sensor}\left({\widehat{T}}_{sensor}\right)-{\widehat{L}}_{up}\left(\widehat{h}\right)-\left(1-\widehat{\epsilon }\right)\left[\widehat{d}· {\widehat{\omega }}_{V/RE}·{\widehat{L}}_{\tau ,V/RE}\left(\widehat{h},T_{V/RE}\right)+\widehat{d}·{\sum }_{i=1}^{10}{\widehat{\omega }}_{sky,i}·L_{\tau ,sky,i}\left(\widehat{h}\right)+(1-\widehat{d})·{\widehat{L}}_{\tau ,spec}\left(\widehat{h},\widehat{j}\right)\right]}{\widehat{\epsilon }+{\widehat{d}·\widehat{\omega }}_x·\left(1-\widehat{\epsilon }\right) }}$$

(14)

However, a second solution for ${\widehat{L}}_x$ has to be considered, as according to Equation (13), an additional instance of ${\widehat{L}}_x$ can arise in Equation (7). The case is relevant if specular reflection is considered ($d>0$) and the specular reflection originates from the local urban environment (j = 0). In this case, the solution yields the followibf:

$${\widehat{L}}_{\tau ,x}\left({\widehat{T}}_x,\widehat{h}\right)={\displaystyle \frac{{\widehat{L}}_{sensor}\left({\widehat{T}}_{sensor}\right)-{\widehat{L}}_{up}\left(\widehat{h}\right)-\left(1-\widehat{\epsilon }\right)\left[\widehat{d}·{\widehat{\omega }}_{V/RE}·{\widehat{L}}_{\tau ,V/RE}\left(\widehat{h},T_{V/RE}\right)+\widehat{d}·{\sum }_{i=1}^{10}{\widehat{\omega }}_{sky,i}·L_{\tau ,sky,i}\left(\widehat{h}\right)\right]}{\widehat{\epsilon }+\left(1-\widehat{\epsilon }\right) [{\widehat{d}·\widehat{\omega }}_x+\left(1-\widehat{d}\right)]}}$$

(15)

Depending on the case, Equations (14) or (15) can finally be used to determine the unknown surface temperature ${\widehat{T}}_x$. For this purpose, the spectral integration of $L_x$, Equation (10), has to be inverted:

$${T_x=\left[L_x\right]}^{-1}\left(L_x,h\right)$$

(16)

As the inversion cannot be carried out in an analytic form, Equation (16) is solved numerically. This can be performed either by creating a look-up table with reasonable resolution for appropriate ranges for $L_x$ and $h$ (see, e.g., [4]), or as in the present paper, by fitting a two-dimensional polynomial to discrete evaluations of the function.

2.4. Preprocessing of Digital Surface Model

2.4.1. Removal of Overhead Artefacts

In order to perform the described sampling algorithm, a clean and highly resolved digital surface model of the survey area is required. However, high-resolution DSMs based on laser-scanning or photogrammetry may contain undesired details, like overhead lines of streetcars, traffic lights or power cables. Meshing algorithms generally tend to exaggerate the size of these thin elements. The resulting extended objects within the DSM are detrimental to the sampling algorithm and should be removed before the sampling process. This can be achieved by either using mesh filtering algorithms or by exploiting the information in additional raster or vector layers covering the survey area. The applied method, however, should only remove undesired elements like overhead wiring of streetcars while preserving any vegetation surfaces, buildings and other relevant surface features (e.g., bridges). In this work, the overhead wiring is eliminated by intersecting a digital terrain model (DTM) with a street vector layer and mapping the DTM street elevations onto the DSM. In order to identify and preserve roadside vegetation and buildings, a hyperspectral classification and building roof cadaster are used.

2.4.2. Determination of Slope Angles

The local surface orientation required for the determination of the incident radiation is calculated using Horn’s method (see [13]). This algorithm performs well compared to other slope calculation methods [14] and provides suitable slope estimations within the context of the present application. For each evaluation point, the surface slope angles are determined in the east-west direction (${\alpha }_x$) as well as north-south direction (${\alpha }_y$), based on the preprocessed DSM at a spatial resolution of 1 m. The slope values for a given DSM pixel are calculated from a three-by-three neighborhood, where direct neighbors are given more weight than diagonal ones. For further information regarding the method and its implementation, refer to [13,15].

2.5. Reflection and Emission of Thermal Radiation

A key feature of the method is the detailed characterization, not only of the surface but also of the environment surrounding it. The environment is the origin of the thermal radiation component that is reflected on the surface and detected by the sensor along with the emission of the surface. Since the method’s objective is to determine the surface temperature, the reflected thermal radiation can be viewed as a disturbance that needs to be compensated for, i.e., subtracted from the detected signal. In order to do this, the reflected radiance needs to be quantified as accurately as possible.

While Fresnel’s equations and the law of reflection are generally valid for thermal radiation, significant specular (i.e., mirror-like) reflection is rarely observed in urban environments. The reason for this is the roughness of the surface. Almost all natural surfaces and most artificial surfaces exhibit a significant level of roughness. If the scale of this roughness is in the magnitude of the wavelength of the incident radiation, or if the surface is constituted of locally flat microfacets that are randomly tilted from the macro surface, the reflected radiation will be diffused. An essential model used for modelling this behavior is Lambertian reflectance. It represents the case in which the reflectance of a surface is constant for any given incidence and reflectance angle, implying that objects with Lambertian surfaces appear equally bright from all viewing angles (see, e.g., Section 4.3.3 of [16]). This is a good approximation for many surfaces and allows a significant level of simplification that is required to model optical scattering efficiently, as applied in this method.

In the model, the radiant intensity of the reflected radiation obeys Lambert’s cosine law:

$$I\left(\alpha \right)=I(0)\mathit{cos}\left(\alpha \right)=I(0)·\left(\overrightarrow{n}·{\overrightarrow{d}}_o\right)$$

(17)

$I\left(\alpha \right)$ … Radiant intensity [W/sr] at angle$\alpha \left[rad\right]$	$\overrightarrow{n}$ … Surface normal vector
$I\left(0\right)$ … Radiant intensity in surface normal direction	${\overrightarrow{d}}_o$ … Outgoing direction (normalized)

In the case of ideal specular reflection, any incident radiation is reflected according to the law of reflection, where the reflectance angle equals the incidence angle. By utilizing vector notation, the law of reflectance can be expressed as follows:

$${\overrightarrow{d}}_o={\overrightarrow{d}}_i-2·\left({\overrightarrow{d}}_i·\overrightarrow{n}\right)·\overrightarrow{n}$$

(18)

${\overrightarrow{d}}_i, {\overrightarrow{d}}_o$ … Incoming and outgoing direction (normalized)

$\overrightarrow{n}$ … Surface normal vector

As stated, specular reflection is less common, especially for thermal radiation. There are, however, important exceptions to that. For the intended use of the method, these are, in particular, the surfaces of polished metals, glass and still water.

The reflection coefficient can directly be derived from the known emissivity of the surface. According to the principle of energy conservation, the emissivity $\epsilon$, reflectivity $\rho$ and transmissivity $\tau$ of the surface have to add up to unity for any wavelengths $\lambda$ and any incident angle $\theta$:

$$\alpha \left(\theta ,\lambda \right)+\rho \left(\theta ,\lambda \right)+\tau \left(\theta ,\lambda \right)=1$$

(19)

Considering that most materials absorb strongly in the thermal radiation spectrum ($\tau \left(\theta ,\lambda \right)=0$), and by applying Kirchhoff’s law ($\alpha \left(\theta ,\lambda \right)=\epsilon \left(\theta ,\lambda \right)$), the reflectivity of the surface can be determined by the following:

$$\rho \left(\theta ,\lambda \right)=1-\epsilon \left(\theta ,\lambda \right)$$

(20)

This, in turn, means that assuming a constant reflectance for the Lambertian reflectance model implies that the surface’s emissivity is constant for all angles. This means that the “constant brightness for all viewing angles” statement linked to the Lambertian reflectance does not only apply to the reflection of the surface but also to its emission.

In reality, the reflectivity of flat surfaces increases for higher incidence angles, reaching a theoretical value of 1 for grazing incidence (90°). This implies a vanishing emissivity for grazing angles. However, the reflectivity and emissivity of most surfaces show little change up to approximately 60°, so the Lambertian model is still a good approximation, particularly for rough surfaces. Figure 6 illustrates the angular dependence of the reflectivity for an ideally flat float glass surface (e.g., a window). The values are achieved by solving Fresnel’s equations for the thermal radiation spectrum (see [17]). Figure 7 shows the directional dependence for the emissivity and radiant intensity of the same surface vs. the Lambert model values in a polar plot representation. As can be seen, the true glass emissivity reaches a value of 0 for 90° (grazing incidence), while the Lambert model emissivity remains at a constant value for all angles. The resulting difference regarding the total radiant intensity is low; however, as the cosine law significantly reduces the radiant intensity for high angles, the differences in emissivity for low incidence angles are small. The depictions and calculations are taken from [17].

2.6. Sampling of Environment

In order to determine the visible classes and their impact, a Monte Carlo sampling algorithm is carried out for each evaluation point (pixel) individually. The approach to apply Monte Carlo sampling for the determination of the incident thermal radiation is based on an approach developed by one of the authors [16], where Monte Carlo sampling methods are used to model heat flux induced by solar radiation. The approach has already been applied for sampling thermal radiation (see [18]). The algorithm is used to determine the incident radiation by casting many rays from the evaluation point into random directions and determining the surface classes that are hit by the rays. There are two main advantages of this method: first, the generation of the rays, as well as the collision point detections, can be performed in a numerically efficient way, where vector algebra is applied, and trigonometric functions are avoided. Second, the random directions used for sampling are generated in a way that ensures that the incident radiance of each direction contributes equally to the total incident radiation, i.e., the same constant weighting factor applies. This is the most efficient approach, as any sampling with non-constant weighting factors implies that a larger number of samples is required to achieve the same accuracy. In order to achieve this, the probability distribution function (PDF) of sampling directions has to be chosen in a way that it represents the underlying physics. In the case of sampling the hemispherical incidence to a Lambert surface, this means that the probability density of the angles to the surface normal $\theta$ of the generated sampling rays (representing the incidence angles) has to decrease proportionally to $\mathit{cos}\theta$. As the azimuthal angles $\phi$ show no preferential orientation, a uniform distribution is applied for these angles. In the implementation, a rejection method is used for this purpose to, again, avoid the computationally expensive use of trigonometric functions. The actual implementation of the approach is presented as pseudocode in Listing 1. For more details regarding the principles.

Listing 1. Implementation core of the algorithm used for the numerical generation of sampling direction vectors based on the Lambertian surface model, refer to Section 4.11 of [16].

For the case of specular reflection, additional sampling is performed based on the law of reflection, Equation (18). Applying the principle of reversibility of the light path, the negative z-vector, representing the nadir direction from the airborne sensor to the ground, is used as the incoming direction ${\overrightarrow{d}}_i$, while the outgoing direction ${\overrightarrow{d}}_o$ provides the desired sampling direction. The, in reality, changing, slightly non-vertical viewing direction across the sensor array is currently disregarded but could be considered in future versions of the algorithm.

The sampling algorithms are then used to determine the dimensionless, fractional coefficients referred to as view factors ($\widehat{\omega }$) of the different classes for each evaluation point (pixel). Based on N samples cast from a location, the view factors can simply be determined by the following:

$${\widehat{\omega }}_x={\displaystyle \frac{N_x}{N}}, {\widehat{\omega }}_{V/RE}={\displaystyle \frac{N_{V/RE}}{N}}, {\widehat{\omega }}_{sky,i}={\displaystyle \frac{N_{sky,i}}{N}}$$

(21)

where $N_x$, $N_{V/RE}$ and $N_{sky,i}$ are the numbers of samples that hit the classes ‘urban surface’, ‘vegetation or remote environment’, and the ten angular sky segments, respectively. Note that the view factors applied here are not based on the standard definition, as they are not equal to the viewed solid angles. Instead, they represent the viewed solid angles scaled by the cosine of the angle at which the related surfaces are visible. However, no explicit and potentially elaborate determination of the solid angles and viewing angles is required, as this is implicitly performed by the Monte Carlo sampling algorithm as described above. Hence, after the sampling is performed, the task of calculating the view factors is reduced to a mere counting of the samples hitting specific surface classes. Note also that even though the introduced view factors represent solid angles that are scaled, based on their definition, all view factors still add up to unity:

$${\widehat{\omega }}_x+{\widehat{\omega }}_{V/RE}+\sum _{i=1}^{10}{\widehat{\omega }}_{sky,i}=1$$

(22)

The determination of the hit class based on a given sampling ray is performed by applying a stepping algorithm. While vector algebra and spatial partitioning (see Section 7.4 of [16]) can be used to solve this task for a triangulated terrain mesh, the data will generally be provided in the form of a digital surface map (DSM). In this case, it is more efficient to discretely step along the path of the sampling rays and compare the current $z$-coordinate with the local elevation in the raster data.

$${\overrightarrow{p}}_0=\left(\begin{array}{c}x\ast sc\\ y\ast sc\\ z(x,y)\end{array}\right)$$

(23)

$${\overrightarrow{p}}_i={\overrightarrow{p}}_{i-1}+sc·\overrightarrow{V}$$

(24)

$$\mathrm{surface} \mathrm{collision}: \overrightarrow{p}.z<z(\left[\overrightarrow{p}.x/sc\right],\left[\overrightarrow{p}.y/sc\right])$$

(25)

$x,y$ … Discrete raster (pixel) coordinates (integer)

$z(x,y)$ … Raster elevation value at pixel (x,y)

$sc$ … Scaling factor [m]

${\overrightarrow{p}}_i$ … Location vector on ray after i steps [m]

$\overrightarrow{V}$ … Sampling direction

2.7. Considered Material Classes and Applied Emissivity Values

Another project group performed the classification of the urban surface. Considerations regarding the method and accuracy are not part of this work, as it is considered input data for the method. The classification is based on a multi-level machine learning approach using hyperspectral images. A hyperspectral camera of type Aisa-FENIX 384 [19] with 364 bands in the VNIR and SWIR (0.4–2.5 μm) was used. More details regarding the survey and classification are provided in [20].

The classification provided 18 different classes to characterize the land cover in the survey area. The classification includes a generic “shadow” class for shaded regions that cannot be classified further. Consequently, an average emissivity value was assigned to this class.

The level of detail and accuracy of the classification determines the overall quality of the results of the method presented in this publication. However, even if no classification data are available and uniform constant emissivity is assigned to the entire area, the processed temperatures will better estimate the true surface temperatures than the unprocessed sensor data, i.e., brightness temperatures. The classification of urban surfaces is considered input data for the method; therefore, no further considerations regarding its acquisition and quality shall be provided here.

In line with the Lambertian surface assumption, the hemispherical emissivity values were applied for most surfaces. The normal emissivity value was used for the predominantly flat water and glass surfaces only. The normal emissivity is a good approximation for incidence angles less than 30 degrees. The values for these two surfaces were specifically derived for the used sensor based on the material’s infrared spectrum by applying a method described in [17].

Table 1. Overview of applied emissivity values and relevant references.

Surface Cover Type	${\overline{\mathit{\epsilon }}}_{\mathit{e}\mathit{f}\mathit{f}}$	Reference
Artificial turf	0.95	Yaghoobian et al. (2010) [21]
Asphalt/concrete	0.95	ECOSTRESS [22]
Bare soil	0.97	Lillesand and Kiefer (1994) [23], adapted
Black roof	0.92	ECOSTRESS [22]
Extensive green roof	0.95	Cascone et al. (2019) [24]
Glass/PV/solar collector	0.878 (N)	Rüdisser et al. (2022) [17], ECOSTRESS [22]
Grass/low vegetation	0.97	ECOSTRESS [22]
Grey roof/gravel	0.93	Kern (2015) [25]
Grey roof/shingles	0.93	Kern (2015) [25]
Metal roof	0.75	Bitelli et al. (2015) [4], adapted for study area
Railway tracks (gravel)	0.93	Kern (2015) [25]
Red tiled roof	0.93	Kern (2015) [25]
Sand	0.91	ECOSTRESS [22]
Shadow	0.95	assumed equal to asphalt/concrete
Sports ground	0.97	Hang et al. (2013) [26]
Trees/forest	0.97	ECOSTRESS [22]
Water	0.984 (N)	Rüdisser et al. (2022) [17], ECOSTRESS [22]
White roof	0.90	Kotthaus et al. (2014) [27], Parker et al. (2000) [28] adapted

Table 1. Overview of applied emissivity values and relevant references.

Surface Cover Type	${\overline{\mathit{\epsilon }}}_{\mathit{e}\mathit{f}\mathit{f}}$	Reference
Artificial turf	0.95	Yaghoobian et al. (2010) [21]
Asphalt/concrete	0.95	ECOSTRESS [22]
Bare soil	0.97	Lillesand and Kiefer (1994) [23], adapted
Black roof	0.92	ECOSTRESS [22]
Extensive green roof	0.95	Cascone et al. (2019) [24]
Glass/PV/solar collector	0.878 (N)	Rüdisser et al. (2022) [17], ECOSTRESS [22]
Grass/low vegetation	0.97	ECOSTRESS [22]
Grey roof/gravel	0.93	Kern (2015) [25]
Grey roof/shingles	0.93	Kern (2015) [25]
Metal roof	0.75	Bitelli et al. (2015) [4], adapted for study area
Railway tracks (gravel)	0.93	Kern (2015) [25]
Red tiled roof	0.93	Kern (2015) [25]
Sand	0.91	ECOSTRESS [22]
Shadow	0.95	assumed equal to asphalt/concrete
Sports ground	0.97	Hang et al. (2013) [26]
Trees/forest	0.97	ECOSTRESS [22]
Water	0.984 (N)	Rüdisser et al. (2022) [17], ECOSTRESS [22]
White roof	0.90	Kotthaus et al. (2014) [27], Parker et al. (2000) [28] adapted

The ECOSTRESS caption in Table 1 references the ECOSTRESS spectral library [22] that includes data provided by NASA’s Jet Propulsion Lab, Johns Hopkins University and United States Geological Survey. Since this database provides spectrally resolved emissivity values for a large number of classes, the spectra were used to calculate an effective emissivity ${\overline{\epsilon }}_{eff}$ for the present survey that considers the spectral sensitivity of the sensor, as well as a mean brightness temperature $T$. A weighted average following Equation (26), or, respectively, the corresponding discrete summation, was applied to determine the effective emissivities relevant for the specific sensor and temperature range. Since the dependence on $T$ is weak in the relevant temperature range, a mean value of 300 K was used for all emissivity evaluations.

$${\overline{\epsilon }}_{eff}\left(T\right)={\displaystyle \frac{{\int }_{\lambda min}^{\lambda max}S\left(\lambda \right) M\left(\lambda ,T\right) \epsilon \left(\lambda \right) d\lambda }{{\int }_{\lambda min}^{\lambda max}M\left(\lambda ,T\right) \epsilon \left(\lambda \right) d\lambda }}$$

(26)

$S\left(\lambda \right)$	…	Spectral sensitivity of sensor
$M\left(\lambda ,T\right)$	…	Planck’s spectral radiant exitance [W·m−2·µm−1]
$\epsilon \left(\lambda \right)$	…	Spectral emissivity

In Figure 8, the reflectivities $\rho$ of all surface classes are illustrated. Based on Equation (20), it can simply by derived by $\rho =1-\epsilon$. It can be seen that most classes have a low reflectivity for thermal radiation ranging up from 2% (water) to 10%. The notable exceptions are glass and particularly metallic surfaces. Since glass surfaces are mostly flat and the material composition of float glass used for building applications usually provides a predictable infrared spectrum (see [17,29,30]), the applied emissivity value for glass should be valid for most urban glass surfaces, as long as they are not covered by significant amounts of dust or dirt.

The situation is, however, a lot more complex for metallic surfaces. Firstly, metallic surfaces can be composed of varying types of metals. Beyond that, and usually even more relevant, metallic surfaces in outdoor environments rarely exhibit the properties of polished pure materials; instead, the interaction of the surface with thermal radiation is determined by its roughness, finish type, possible coatings, oxidation or weathering. Consequently, there is a wide range of potential emissivity values and reflective characteristics relevant for metallic surfaces. The topography of rough surfaces usually comprises surface structures of scales that are in the range of the wavelengths of the relevant thermal radiation. This usually leads to a reduced reflectivity and, in turn, higher emissivity of the surface (see, e.g., [31])

In the present work, the available hyperspectral classification did not allow for a more detailed modelling approach for metallic surfaces. Following a heuristic approach, an emissivity value of 0.75 and a diffuseness value of 0.1 were determined to provide suitable average results. However, this constant approach leads to a significant over- and underestimation of the reflected thermal radiation for some surfaces. The underestimation of the reflection for horizontally, or near horizontally, oriented smooth metallic surfaces with low emissivity values implies that the determined surface temperatures are significantly too low. The effect is commonly known in thermography when the directly detected brightness temperatures of metallic roofs can range below freezing point even on summer days, as the roofs predominantly reflect the downwelling radiation from the sky.

On the contrary, the emissivity value of water is very high if the camera’s relevant spectral range is considered. Hence, water surfaces can be used to validate or calibrate the atmospheric transfer models, as the impact of reflected thermal radiation is very limited. Beyond that, the orientation of a still water surface is well determined.

2.8. Validation Approach

The method proposed here was developed and implemented well after the survey flights were performed. At the time of the survey, deriving surface temperatures from the thermography data was not planned. Hence, no ground measurements specifically designed to validate the method were performed. In order to still evaluate the results, suitable data sources that continuously record temperatures were identified, and data from the most suitable stations were gathered and processed. The available data mostly comprised air temperature measurements but also water temperatures of the river Mur. Additionally, two suitable contact temperature measurements performed by other project partners during the survey could be used.

2.8.1. Considerations Regarding Water Temperature Validation Measurements

As mentioned in Section 2.7, temperatures of water surfaces provide a good calibration or validation source for the atmospheric model, as the influence of reflected thermal radiation is limited due to the high emissivity of water surfaces. However, it has to be considered that, due to the high absorption in the relevant wavelength range, the radiometric temperature measurements reflect the temperature of the first few micrometers of the water surface, which is referred to as skin temperature. This skin temperature can significantly deviate from the bulk water temperature due to solar exposition and evaporative or thermal cooling (see, e.g., [32,33]). Additional stratification of the upper water temperatures can be expected on still surfaces (i.e., lakes).

2.8.2. Considerations Regarding Contact Temperature Measurements

Contact temperature measurements, usually performed with temperature probes that hold a thermoelement, are valuable tools for validating or calibrating radiometric temperatures. For this purpose, special attention should be given to accurately geo-referenced documentation of the measurement location, a well-established contact between the probe and the surface, and a short measurement time to limit any disturbances of the measurement, e.g., by shading. Further, it has to be ensured that each measurement reflects the average temperature of the potentially varying surface temperature in the immediate vicinity. The ground-based contact temperature measurements in the application covered here were carried out by another measurement team during the period of the aerial survey. The measurements were performed using a handheld contact thermometer of type Voltcraft Multi-Thermometer DT-300 with a specified maximum absolute error of 1 K.

2.8.3. Considerations Regarding Validation Against Air Temperatures, Leaf Temperatures and Backside Foliage Approach

In order to perform a larger number of validations, available air temperature recordings of various stations were compared against the determined temperatures of vegetation surfaces. For this purpose, a literature review was carried out regarding the relation of leaf surface temperature to air temperature. The findings in the reviewed publications [34,35,36,37,38] do not present a clear pattern. While the widely applied limited homeothermy hypothesis proposed by Mahan und Upchurch [39] and refined by Michaletz et al. [40] assumes that the leaf temperatures fall below air temperature for higher air temperatures, recent studies seem to contradict this (see, e.g., [9,34]).

Surprisingly, none of the mentioned studies considered the influence of reflected thermal radiation, even though all mentioned publications are based on radiometric, i.e., thermographic, measurement principles. In the mentioned studies, varying temperatures depending on the tree’s location and the vertical position of the measurement relative to the tree’s canopy were observed. Richter et al. [36] additionally observed a significant absolute offset between radiometric LST measurements performed with a low-flying gyrocopter vs. Landsat data. All of these observations might well be caused or influenced by a variation of the reflected thermal radiation, depending on the local environment, sky condition, spectral and directional emissivity value and viewing angles.

Regarding the varying and even contradictory findings in the reviewed publications and considering the typical heat transfer coefficients and the relatively large surface ratio of foliage, we assume that the surface temperatures of leaves with limited solar exposition might not deviate significantly from the surrounding air temperature (if determined correctly). The following validations are based on this hypothesis. We are aware that this approach and, therefore, the related validation results are debatable. However, we think that the application of the proposed backside foliage method contributes to the findings of this study.

Based on the influencing factors, several considerations regarding the ideal measurement location for determining near-air temperatures must be taken into account (see Figure 9). Firstly, direct exposure to sunlight should be avoided (1). The measurement location should be offset to the tree’s center, ideally in a sparsely dense region (2). This should ensure that the measurements reflect an integrated value along an extended path through the tree and are dominated by foliage surfaces rather than the trunk or branch surfaces. Ideally, the thermal radiation reflected by the target originates again from vegetation surfaces (3). In particular, significant exposure to the sky should be avoided. Finally, an open region around the tree should ensure leaves are exposed to the “free” air temperature (4), as air temperatures in dense vegetation can significantly deviate. While some of these considerations represent conflicting requirements, it was still possible to identify suitable measurement locations for the validations that comply largely with the proposed approach.

The considerations above are relevant for determining leaf temperatures reflecting the meteorological air temperatures, which are commonly measured at a location 2 m above ground level. However, two local measurement stations of the national meteorological agency, GeoSphere Austria, additionally provide air temperatures measured at 5 cm above ground level. Especially at night, these temperatures can deviate significantly from the ones measured at 2 m. In order to include this information in the validation, the surface temperature of the grass was determined for the specific locations. For this evaluation, the solar exposure has to be taken into account.

3. Results—Application of the Method

3.1. Survey Parameters

Two identical survey flight paths covering the entire area of the city of Graz were carried out on September 9th and 10th 2021. The flight carried out on the 9th is referred to as “12 UTC” and was performed in the period of 13:10 to 15:04 local time. The purpose of this flight was to capture the heated urban environment. The second flight on the following day is further referred to as “03 UTC” and was performed from 4:10 to 6:06 local time. Its purpose was to capture the coldest period of the day. The local sunrise on the survey day occurred at 6:27, while the civil dawn time was 5:57. A dual camera of type DigiTHERM-1024 [41] was used for the aerial survey. The thermal camera integrated into the dual-camera system was of the InfraTec HD 800 type [42]. The flight altitude of the surveys was 1675 m MSL.

A high-pressure system with clear skies dominated the local meteorological situation. In the night survey, a ground inversion with a significant temperature inversion was detected, with the highest air temperatures observed at a station located at 710 m MSL. This weather situation is frequently observed in Graz [43] and is mainly caused by the basin-like topography formed by the hills and mountains around the city. Air temperatures of most stations stayed below 25 °C throughout the day. A clear sky characterized the night and day survey, with a solar irradiance of approximately 820 W/m² of direct normal irradiance and 730 W/m² of global horizontal irradiance at the time of the day survey.

3.2. Atmospheric Modelling

As described in Section 2.2, data from four different sources were merged to model the relevant atmospheric profiles. Station data were used to model the humidity and temperature at ground level, local forecast and radiosonde data were used to model the higher layers up to 25 km altitude, and finally, the generic profile ‘mid-latitude summer’ was included in the MODTRAN tool [11,12] was used to model the high altitude layers ranging up to 100 km. The ‘mid-latitude summer’ profile was also used to model the entire concentration profiles of the further greenhouse gas components. The concentration of CO₂ was upscaled to match the measurements of the Austrian Sonnblick Observatory [44] recordings for that date. Figure 10 and Figure 11 illustrate the gas concentrations, atmospheric pressures and air temperatures as applied. A log–log representation is chosen for the gas concentration profiles to provide a comprehensive overview. However, the relevance of the concentrations, and therefore the need for accurate modelling, decreases along with atmospheric pressure, as the absolute concentration is relevant for the radiative interactions. In order to illustrate this, the absolute concentrations of the water vapor for both surveys are depicted in Figure 12. Additionally, the cumulative absolute humidities, representing vertical columns, are shown with the dashed graphs. Roughly 70% of the water molecules are located below the flight altitude of 1675 m MSL, whereas more than 90% are concentrated below 4000 m altitude. In order to consider additional scattering on aerosols, the generic “urban” model of MODTRAN was selected. The related visibility parameter was set to 100 km.

3.3. Processing of Raster Data

The following raster datasets were used in the processing (each given at a spatial resolution of 1 m)

• Photogrammetry-based DSM (digital surface map covering the city of Graz, 2021);
• Laser-scanner-based DSM (digital surface map covering the areas surrounding Graz, acquisition period 2008–2014);
• Laser-scanner-based DTM (digital terrain map covering the city of Graz, 2018);
• Hyperspectral land cover classification (covering the city of Graz, 2021).

In addition, ancillary vector data of the survey area were utilized for preprocessing the DSMs:

5.

Shapefile of roads within Graz (2021);

6.

Roof cadastre of Graz (2019).

The photogrammetry-based DSM (1) was recorded concurrently with the 12 UTC flight and covers the same area as the thermal imagery, i.e., the entire city area of Graz. However, information on the surface elevations beyond the city area of Graz is required to consider the remote environment, i.e., mountains and hills, in the sampling algorithm. For this purpose, a laser-scanner-based DSM (2) [45] is used. It extends from the Schöckl mountain in the north to Graz airport in the south, covering a total area of approximately 500 square kilometers. The topographic information contained in the DSMs (1) and (2) was merged to obtain a combined DSM that covers the city area of Graz and its surroundings with high resolution.

As mentioned in Section 2.4, the combined DSM has to be filtered, as it contains overhead wirings, mostly of tram lines, which would interfere with the sampling algorithm. A preprocessing algorithm was designed and implemented to eliminate the wiring from the DSM, utilizing the information provided by components (3) to (6). As the overhead wiring is mainly located above road surfaces, the proposed filtering process is restricted to road areas within the city, which are identified using component (5). The filtering now relies on the fact that the street elevations within the DSM and the DTM (3) are largely identical unless surface features like overhead wiring or cars are present within the DSM.

In order to remove the overhead wiring and other unwanted surface features from the road areas, the street elevations in the DSM are replaced with the corresponding elevation values taken from the DTM. However, this process would not only eliminate the overhead wiring from the DSM but also remove roadside vegetation and overlapping roof elements. In order to prevent this, the ancillary information provided by components (4) and (6) is used. Based on the available classification (4), vegetation areas are identified, vectorized, and preserved. Similarly, component (6) is used to preserve overlapping building elements, primarily roofs. In addition, bridges were manually outlined and excluded from the filtering procedure. Finally, the filtering process is carried out to effectively eliminate unwanted surface features while preserving roadside vegetation and overlapping roof elements. Figure 13 illustrates the filtering performance based on a detail that contains a significant amount of overhead wiring. More details on the DSM preparation are available in [15].

The filtered DSM data are further used to determine the slope angles by applying the algorithm as described in Section 2.4.2. An exemplary detail of the two resulting slope maps is depicted in Figure 14. Next, the sampling process (see Section 2.6) is carried out based on the filtered DSM and the derived slope maps. The sampling algorithm results in 13 raster layers containing the scalar view-factor values in a two-dimensional form (see Figure 15). Finally, all components required to determine the surface temperature are available. This final step is performed by deriving the radiance values ${\widehat{L}}_{\tau ,x}\left({\widehat{T}}_x,\widehat{h}\right)$ based on Equation (14) or, respectively, Equation (15), and solving Equation (16) to determine the surface temperatures.

Figure 16 and Figure 17 provide an overview of the entire processed survey area. While detailed validations of the derived surface temperature are included in the next chapter, it is immediately apparent that the determined surface temperature maps generally show higher temperatures but also provide better contrast, i.e., a higher temperature range. This result aligns with the applied models, as the atmospheric influences, on the one hand, and the reflection of thermal radiation, on the other hand, will generally decrease the range of temperatures detected at the sensor.

3.4. Validation Measurements

3.4.1. Comparison with Water Temperatures—Mur River

The temperature of the river Mur running through Graz is continuously measured by the local hydrographic service of Styria [46]. The recorded temperatures at the time of the surveys (${\widehat{T}}_{water}$) are compared against the detected temperatures of the water surfaces in the model considering the specific measurement location (${\widehat{T}}_x$ and ${\widehat{T}}_{sensor}$). The location of the measurement is depicted in Figure 18. An overview of the relevant temperatures is provided in

Table 2. Probe measured (${\widehat{T}}_{water}$), apparent temperature at sensor (${\widehat{T}}_{sensor}$) and determined temperature (${\widehat{T}}_x$) for four different measurement times.

Surface Cover Type	${\widehat{\mathit{T}}}_{\mathit{w}\mathit{a}\mathit{t}\mathit{e}\mathit{r}}$	${\widehat{\mathit{T}}}_{\mathit{s}\mathit{e}\mathit{n}\mathit{s}\mathit{o}\mathit{r}}$	${\widehat{\mathit{T}}}_{\mathit{x}}$	${\widehat{\mathit{T}}}_{\mathit{s}\mathit{e}\mathit{n}\mathit{s}\mathit{o}\mathit{r}}-{\widehat{\mathit{T}}}_{\mathit{w}\mathit{a}\mathit{t}\mathit{e}\mathit{r}}$	${\widehat{\mathit{T}}}_{\mathit{x}}-{\widehat{\mathit{T}}}_{\mathit{w}\mathit{a}\mathit{t}\mathit{e}\mathit{r}}$
survey 8 September 2021 14:30 UTC	16.7	14.54	17.55	−2.16	0.85
survey 9.September 2021 03 UTC	14.8	12.22	14.51	−2.58	−0.29
survey 9 September 2021 12 UTC	16.4	15.77	17.94	−0.63	1.54
survey 10 September 2021 03 UTC	14.6	13.17	14.25	−1.43	−0.35

. The values were determined by averaging water temperatures in an area of approximately 100 m² around the sensor position. In addition to the temperature evaluations of the two surveys covered in this publication, two evaluations of surveys of a larger area with lower spatial resolution performed on the preceding days, but not otherwise covered here, are included in this comparison.

3.4.2. Comparison with Contact Temperature—Lendplatz Pavement

The contact temperature measurement was performed on the concrete paving located on the Lendplatz square in the city center. It serves as the reference temperature for the day survey. The location of the measurement and evaluation profile path, as well as the evaluation results, are depicted in Figure 19. The ground measurement was carried out at 13:30 local time during the period of the aerial survey.

3.4.3. Comparison with Contact Temperature—Färberplatz Pavement

The contact temperature measurement, which serves as a reference temperature for the day survey, was performed on a shaded part of the cobblestone paved Färberplatz square, also located in the center of the city. The location of the measurement and evaluation profile path, as well as the evaluation results, are depicted in Figure 20. The ground measurement was carried out at 14:15 local time within the period of the aerial survey.

3.4.4. Comparison with Air/Vegetation Temperatures—Schlossberg

The station is operated by the local Styrian government [47] and is located on the central hill (Schlossberg) in the city at an elevation of approximately 460 m MSL. The shadow side of a row of trees at a close distance to the measurement station was evaluated. The location of the measurement and evaluation profile path, as well as the evaluation results, are depicted in Figure 21.

3.4.5. Comparison with Air/Vegetation Temperatures—Oeverseepark

The station is operated by the local Styrian government [47] and is located close to the city center at an elevation of 348 m MSL. Again, the shadow side of a row of trees at a close distance to the measurement station was evaluated. The location of the measurement and evaluation profile path, as well as the evaluation results, are depicted in Figure 22.

3.4.6. Comparison with Air/Vegetation Temperatures—Lustbuehel

The station is operated by the local Styrian government [47] and is located near an observatory on a hill at the southeastern boundary of the city at an elevation of 480 m MSL. The shadow side of a row of trees at a close distance to the measurement station was evaluated. The location of the measurement and evaluation profile path, as well as the evaluation results, are depicted in Figure 23.

3.4.7. Comparison with Air/Vegetation Temperatures—Plabutsch

The station is operated by the local Styrian government [47] and is located near the highest point in the survey area on the Plabutsch mountain at an elevation of approximately 750 m MSL. The shadow side of the edge of a forest at a close distance to the measurement station was evaluated. The location of the measurement and evaluation profile path, as well as the evaluation results, are depicted in Figure 24.

3.4.8. Comparison with Air/Vegetation Temperatures—Mariatrost

The station is operated by the private company Andreas Pilz Umweltmesstechnik. It is located in a valley spreading to the northeast of the city boundary. The location is known for exhibiting some of the coldest nighttime temperatures due to the frequent formation of significant night inversions in the basin-like topography. The station is located at an elevation of 424 m MSL. The shadow side of a row of trees at a close distance to the measurement station was evaluated. The location of the measurement and evaluation profile path, as well as the evaluation results, are depicted in Figure 25.

3.4.9. Comparison with Air/Vegetation Temperatures at 2 m and 5 cm—Graz University

The station is operated by the national meteorological institute GeoSphere Austria [48]. The station provides the primary official temperature readings for the city of Graz and is located on the university campus near the city center. The elevation is 366 m MSL. The measurement station provides data on air temperatures measured at 2 m and 5 cm above ground level. The location of the measurement and evaluation profile path, as well as the evaluation results, are depicted in Figure 26.

In order to determine vegetation temperatures representing the air temperatures at 5 cm above ground level, the grass surface inside the fenced area of the measurement station was determined. Again, the location of this measurement and evaluation profile path, as well as the evaluation results, are depicted in Figure 27.

3.4.10. Comparison with Air/Vegetation Temperatures at 2 m and 5 cm—Graz Strassgang

The station is operated by the national meteorological institute GeoSphere Austria [48]. The station is located on the premises of a public schoolyard in the west of the city. The elevation is 357 m MSL. The measurement station provides air temperatures measured at 2 m and 5 cm above ground level.

In order to determine the 2 m near-air temperatures, the surface temperatures of the foliage on the backside of a tree in close vicinity to the station were evaluated.

Again, an additional air temperature sensor mounted 5 cm above ground level was available, and, again, the surface temperature of the grass near the sensor was determined and compared against the air temperature recordings. The location of both measurement and evaluation profile paths, as well as the evaluation results, are depicted in Figure 28 and Figure 29.

4. Discussion

4.1. Discussion of Validation Results

4.1.1. Water Temperature Comparisons

While the determined water temperatures of the two night surveys show an excellent agreement, well below the sensor’s absolute error of 1 K [41,42], the radiometrically determined temperatures for the two daytime surveys exceed the in-situ measured water temperatures by 0.85 K (first survey) and 1.54 K (second survey), see Table 2. It is likely that this temperature difference can partially be attributed to the heating of the water skin surface of the river and/or the mist above it (see Section 2.8.1). The potential reason is the high solar exposition at that time, which was approximately 457 W/m² of global horizontal irradiance for the first survey, and 732 W/m² for the second survey. The almost linear relationship between the deviation of the radiometrically determined water temperature vs. the contact-measured water temperature (at a lower depth) supports this assumption, see Figure 30. Of course, further investigation is required here. Additional radiometric water temperature validation measurements should be performed locally in a follow-up study using handheld pyrometers.

4.1.2. Contact Temperature Comparisons

For the location Lendplatz (see Section 3.4.2), the determined temperatures ${\widehat{T}}_x$ are on average 7.1 K higher than the detected (uncompensated) temperatures ${\widehat{T}}_{sensor}$. The determined temperatures show an excellent agreement with the locally determined contact temperature of 34.8 °C. At the location Färberplatz (see Section 3.4.3), the determined surface temperatures ${\widehat{T}}_x$ are on average 3.3 K higher than the detected temperature ${\widehat{T}}_{sensor}$, and are, again, in good accordance with the locally determined contact temperature of 23.1 °C.

4.1.3. Air Temperature/Vegetation Temperature Comparisons

• Location Schlossberg (see Section 3.4.4)

In this specific case, the determined foliage surface temperatures for the night survey do not vary significantly from the sensor temperatures and show good agreement with the local air temperature measurements. Following the considerations above, the highest surface temperatures should represent the air temperatures best, as deviations to colder temperatures are likely caused by significant exposition to the cold night sky.

The determined daytime surface temperatures range significantly above the sensed temperature and show good accordance with the measured air temperatures. Contrary to the night temperatures, the minima of the determined surface temperature should best represent the air temperature, as higher temperatures are likely a result of solar exposition.

• Location Oeverseepark (see Section 3.4.5)

The results are comparable to the ones of the Schlossberg station. Again, the minima of the daytime foliage temperatures and the maxima of the nighttime foliage temperatures show a good agreement with the air temperature recorded at that time. Whereby the daytime deviations to higher temperatures might again be attributed to solar exposition and the nighttime deviations to lower temperatures might be attributed to exposure to the night sky

• Location Lustbuehel (see Section 3.4.6)

Again, the minima of the determined daytime foliage temperatures show a good agreement with the reference air temperature recorded at that time. The maxima of the determined nighttime foliage temperatures range approximately 1 K to 2 K below the air temperature recorded at that time. The likely reason for the deviation to colder temperatures is the significant exposition to the night sky amplified by the exposed location on a hilltop.

• Location Plabutsch (see Section 3.4.7)

The minima of the determined daytime foliage temperatures show a good agreement with the air temperature recorded at that time. The maxima of the determined nighttime foliage temperatures range again approximately 1 to 2 K below the recorded air temperatures. The likely reason for the deviation to colder temperatures is, again, the significant exposition to the night sky amplified by the exposed location on a mountaintop. Unlike in the preceding cases, the determined nighttime temperatures range significantly above the detected ones due to changes in atmospheric parameters with increased elevation.

• Location Mariatrost (see Section 3.4.8)

The minima of the determined daytime foliage temperatures range below the recorded air temperature; however, the average determined foliage value is close to it. Due to the different altitude and environment, and unlike in preceding cases, the nighttime surface temperatures range above the recorded air temperature. Also deviating from the previously presented cases, the determined nighttime foliage temperatures now range on average 2.0 K below the temperature detected at the sensor, while the daytime temperatures are offset by 3.3 K to higher temperatures.

• Location Graz University at 2 m level (see Section 3.4.9)

In order to determine the 2 m near-air temperatures, the surface temperatures of the foliage on the backside of a shrub in close vicinity to the station were evaluated.

While the minimum of the daytime surface temperature profile is close to the recorded air temperature (+0.6 K), the average values range significantly above that value (+1.6 K). The determined maximum of the nighttime foliage temperatures range is −1.0 K below the recorded air temperature, while the average values are offset by 1.2 K. The likely reason for both deviations is the high exposure to the sky, leading to elevated temperatures during the day (due to diffuse solar irradiance), and additional cooling during the night. However, no other suitable vegetation with less sky exposure is located near the station.

• Location Graz University at 5 cm level (see Section 3.4.9)

The radiometrically determined grass surface temperatures are exceptionally close to the air temperatures recorded by the station at 5 cm above ground level. Since no relevant topography-based variation along the measurement path for the grass can be expected, the average values along the path are used to determine the grass temperatures.

The determined daytime grass surface temperature is identical (0.0 K) to the recorded air temperature (32.9 °C), while the radiometrically determined average nighttime grass temperature ranges 1.1 K above the recorded air temperature (7.5 °C).

• Location Graz Strassgang at 2 m level (see Section 3.4.10)

While the minima of foliage temperatures determined during daytime show good agreement with the recorded air temperature, the temperatures determined at nighttime are systematically lower by approx. 2 K.

• Location Graz Strassgang at 5 cm level (see Section 3.4.10)

Similar to the comparison result for the 2 m sensor, the temperature determined at nighttime is systematically offset to colder temperatures by 1.3 K. The determined daytime grass temperatures show an excellent agreement with the recorded air temperatures as they are, on average, just 0.1 K above the reference. A potential source for the nighttime deviations of the 5 cm and the 2 m measurements of this location could be the relatively exposed measurement location. Consequently, the surfaces might not be in full equilibrium with the surrounding air body due to higher natural or forced (wind-driven) convection.

4.2. Altitude, Atmosphere and Emissivity Sensitivity Analysis

The results of a brief sensitivity study based on the atmospheric models of both surveys with the assumption of an unobstructed view of the horizon are illustrated in Figure 31. As can be seen, all determined temperatures range significantly above the temperature sensed by the detector ($T_{sensor}$). The parameter with the highest sensitivity is clearly the emissivity of the surface. Even for the case of $\epsilon =1.0$, where the reflection of thermal radiation is irrelevant, the atmospheric transfer leads to a significantly reduced temperature reading, ranging approximately 2–3 K below the surface temperature. The temperature delta between the actual and sensed surface temperature increases considerably for surfaces with lower emissivities. The elevation dependence is clearly visible but decreases for lower emissivity values, as the gradient of transmittance is partly compensated for by the gradient of the reflected downwelling radiation.

The difference between the two surveys is visible but not pronounced, as the two flights were carried out within 15 h. While some variations linked to the diurnal cycle occur in the lower atmospheric layers, the conditions of the higher atmospheric layers largely remain stable.

4.3. Exemplary Analysis—Solar Thermal Field

In order to demonstrate the performance of the algorithm regarding surfaces with lower emissivities, a detail covering a solar thermal collector field in the north of the city is presented (see Figure 32). The emissivity value of float glass of 0.878 [17] was assigned to the surfaces representing the thermal collectors. The relatively low emissivity value implies that the detected radiation comprises 12% of reflected thermal radiation. Due to the orientation of the glass surfaces, this thermal radiation predominantly originates from the sky. Hence, the detected temperature, i.e., apparent surface temperature at the sensor, ranges significantly below the actual surface temperature.

As can be seen in Figure 33 and Figure 34, the detected daytime temperatures of the glass surface range on average 1 K below the air temperature recorded by a nearby weather station (Graz Nord) at that time. After processing, the surface temperatures of the glass covers of the collectors range from 9 K to 13 K above the local air temperature. Considering the high solar exposition at the time of the survey (approx. 820 W/m² of direct normal irradiance and 730 W/m² of global horizontal irradiance), the determined temperature differences are plausible.

A similar effect can be observed during nighttime. While the determined surface temperatures still range below the air temperature, the temperature delta to the air temperature is reduced from −4.5 K (detected temperature) to −2.5 K (determined temperature). Due to the exposition of the glass surface to the night sky, a related radiative cooling below air temperature can be expected.

4.4. Exemplary Analysis—City Center: Mur River and Schlossberg

The example represents a detail of the city center (see Figure 35 and Figure 36). It includes the local river, historic and new buildings, as well as the west side of the forest covering Schlossberg (castle hill). The temperature profile depicted in Figure 37 indicates the temperatures detected and determined along the path depicted as white arrow in the images of Figure 36. As can be seen, all determined temperatures range above the detected temperature. The median value of the temperature delta is 4.8 K. The maximum temperature delta is 33 K and occurs on the metallic roofs of the building close to the river. The WSW-oriented roof reaches a determined temperature of 75 °C, while the ENE-oriented metallic roof reaches 67 °C. The roof pitch angle of both roofs is 19 degrees. Both roof temperatures range approx. 28 K above the detected temperatures. However, the uncertainty of the true emissivity of the metallic surface has to be considered here. The red-tiled roof of the building at the forest’s edge reaches a determined temperature of 56 °C, which is 17 K above the detected temperature of 39 °C. The determined surface temperature of the two intersected roads is approx. 42 °C and ranges approx. 10 K above the detected temperature.

4.5. Exemplary Analysis—“Hauenstein”—Radiation Inversion

In this example, the temperature profile of a hill in the northeast of the survey area (see Figure 38) is analyzed for the night survey “03 UTC”. A significant radiation inversion characterizes the temperature profile. This phenomenon is frequently observed in the region at this time of the year. The topography of Graz leads to the formation of cold-air pools in the main basin, forming the city’s center region, and even more pronounced cold air pockets can be observed in valleys surrounding the city. The reason for this significant temperature inversion is the radiative exchange with the night sky amplified by nocturnal down-slope winds (see, e.g., [49]), especially on clear nights, as observed during the survey. As shown in Figure 39 and Figure 40, the determined temperatures drop from 15–16 °C on the hilltop to approximately 4 °C at the bottom of the valley. Remarkably, the temperatures determined in the valley range considerably below the detected temperatures, while the opposite is true for the temperature on the hilltop. Therefore, the temperature drops of around 8 K observed in the thermography image is increased to approximately 12 K when the surface temperatures are determined. No stationary air temperature measurements were available for the analyzed region. However, mobile air temperature measurements performed with a sensor attached to a car detected a minimum temperature of 4.1 °C on a road near the depicted area at a similar daytime on the day preceding the survey, see [50].

5. Conclusions

A new method for deriving surface temperatures based on aerial thermography and supplemental data sources was developed. The physical fundamentals and models that the method relies on are discussed, and the equations necessary to determine the surface temperatures are derived. The innovative nature of the method is based on the applied three-dimensional sampling process to determine reflected thermal radiation, the angularly resolved model for the downwelling radiation, the altitude-dependent modelling of atmospheric transmittance, downwelling and upwelling radiation, as well as the spectrally resolved consideration of the atmospheric radiative transfer and the sensor sensitivity. The atmospheric temperature, absolute pressure and partial pressure profiles are considered for the six most relevant greenhouse gas components based on forecast data, ground station measurements, radiosonde measurements, and generic concentration profiles (see Figure 10, Figure 11 and Figure 12). The sensitivities of the different components are analyzed (see Figure 4, Figure 5 and Figure 31). The three-dimensional sampling process is used to determine the incident thermal radiation. The sampling is performed based on a detailed digital surface model of the city that is extended with a larger model to consider remote environments at further distances. The emissivities of the surfaces are determined based on a hyperspectral classification map. While emissivity is determined to be the most influential parameter, the emissivity of most urban surfaces lies within a narrow range, reducing the potential errors of the surface temperature determination. However, metallic surfaces pose a significant exception to this. Since they exhibit low emissivities, they reflect considerable amounts of thermal radiation that usually originates predominately from the sky. This, in turn, increases the significance of accurately determining the emissivity in order to derive the correct surface temperature. Since the classification of the metallic surfaces did not allow further differentiation and the effective emissivity of these surfaces relies on many factors like material composition, weathering, roughness, coating and dust, the determined surface temperatures of metals have the potentially highest errors. A more precise determination of the emissivities of metallic surfaces requires further research. A combined evaluation of thermal and hyperspectral imagery potentially allows for a more detailed classification.

The two applications of the method for the city area of Graz proved to provide stable and plausible results. The determined temperatures are, on average, higher than the apparent temperatures of the thermography images. The main reason for this is the detailed consideration of the reflection of downwelling radiation in the model. However, for surfaces with higher emissivities, e.g., natural surfaces, the determined temperatures can also range significantly below the detected temperatures (see, e.g., Figure 40). Hence, the determined surface temperature images generally exhibit a higher contrast as they cover an increased temperature range. The observed loss of contrast in airborne thermography, which is largely compensated for by this method, can be explained by the fact that the atmospheric influences and the reflection of thermal radiation generally imply attenuation and averaging effects that reduce the range of the detected temperature. The considerably increased range of the surface temperatures relative to the detected temperatures at the sensor is demonstrated for the slope of a hill exposed to a nighttime radiation inversion (see Section 4.5). In the example, the determined range of vegetation temperatures spans over 8 K in the apparent temperature image, while it reaches 12 K for the determined surface temperatures (see Figure 39 and Figure 40). The effect is, however, even more pronounced in typical urban environments with artificial surfaces in a three-dimensional configuration (see, e.g., Figure 36 and Figure 37). In particular, the apparent temperatures of roof surfaces are substantially lower than their actual surface temperatures. Due to the mostly lower emissivities and near horizontal orientations of roof surfaces, reflected thermal radiation originating from the sky accounts for a considerable share of the detected signal. In the processed city area, the determined temperatures of roofs ranged up to 30 K above the detected temperature. However, the determination of the exact temperature relies on precise knowledge of the surface’s emissivity, as stated above.

Since the method presented here was developed well after the aerial survey, no explicit validation measurements were carried out at the time of the survey. Still, valuable measurement data recorded at the time of the survey could be identified and exploited. The measurements comprise water temperatures, two contact temperature determinations, as well as several air temperature measurements. A new method to relate the recorded air temperatures to radiometrically determined surface temperatures of vegetation is proposed (“backside foliage method”) and tested within the context of this publication. While the available validation measurements do not allow a precise assessment of the absolute errors involved in the temperature determination, the performed comparisons exhibit a good level of agreement and plausibility. A thorough analysis of parameter sensitivities and error of the temperature determination based on specifically designed ground measurements must be part of future applications of this method.