Time-Continuous Hemispherical Urban Surface Temperatures

Abstract

Traditional methods for remote sensing of urban surface temperatures (T_surf) are subject to a suite of temporal and geometric biases. The effect of these biases on our ability to characterize the true geometric and temporal nature of urban T_surf is currently unknown, but is certainly nontrivial. To quantify and overcome these biases, we present a method to retrieve time-continuous hemispherical radiometric urban T_surf (T_{hem, r}) from broadband upwelling longwave radiation measured via pyrgeometer. By sampling the surface hemispherically, this measure is postulated to be more representative of the complex, three-dimensional structure of the urban surface than those from traditional remote sensors that usually have a narrow nadir or oblique viewing angle. The method uses a sensor view model in conjunction with a radiative transfer code to correct for atmospheric effects in three-dimensions using in situ profiles of air temperature and humidity along with information about surface structure. A practical parameterization is also included. Using the method, an eight-month climatology of T_{hem, r} is retrieved for Basel, Switzerland. Results show the importance of a robust, geometrically representative atmospheric correction routine to remove confounding atmospheric effects and to foster inter-site, inter-method, and inter-instrument comparison. In addition, over a month-long summertime intensive observation period, T_{hem, r} was compared to T_surf retrieved from nadir (T_plan) and complete (T_comp) perspectives of the surface. Large differences were observed between T_comp, T_{hem, r}, and T_plan, with differences between T_plan and T_comp of up to 8 K under clear-sky viewing conditions, which are the cases when satellite-based observations are available. In general, T_{hem, r} provides a better approximation to T_comp than T_plan, particularly under clear-sky conditions. The magnitude of differences in remote sensed T_surf based on sensor-surface-sun geometry varies significantly based on time of day and synoptic conditions and prompts further investigation of methodological and instrument bias in remote sensed urban surface temperature records.

1. Introduction

Thermal infrared (TIR) remote sensing of land surface temperature has emerged as a primary focus in climatology, as researchers seek to better describe spatiotemporal patterns of surface temperature (T_surf) globally and better understand how anthropogenic modification of the Earth’s surface influences land T_surf and impacts climate at various scales. Over the last two decades, application of thermal remote sensing to study surface climates has expanded significantly. Thermal remote sensing of the Earth’s surface has applications over a wide range of disciplines: from informing micro-, urban-, and global-scale climate models, to aiding decision making and mitigation strategies with respect to climate change and the urban heat island effect.

Within urban climatology, a combination of satellite, aerial, and ground-based thermal remote sensors have been integral in elucidating the spatial [1,2], temporal [3,4], and geometric [5] effects of the built environment on land T_surf; in evaluating and partitioning urban surface energy balances [6,7,8] and; in characterizations of the relationship between surface and boundary-layer air temperatures (T_air) [9]. These advances have been aided by substantial improvements in sensor spatial, spectral, and radiometric resolutions, and by the proliferation and availability of both large-scale satellite remote sensing products and low-cost aerial and near-ground thermography. However, in spite of its widespread usage, several questions concerning the use and validity of urban remote thermal remote sensing, first posed in Roth et al. [1], have yet to be sufficiently answered, viz,

• What is the nature of the surface ‘seen’ by a thermal remote sensor?
• How does T_surf observed by a remote sensor relate to the ‘true’ temperature governing the surface-atmosphere interface?

In this paper, we seek to examine question two by introducing and evaluating a method for atmospheric correction of near-ground hemispherical TIR radiation—measured via pyrgeometer—for hemispherical radiometric temperature (T_{hem, r}) retrieval. These measures are common to most urban energy balance assessments because they are made as a part of the net radiation measurement and thus constitute a hitherto untapped method for urban T_surf analysis with a number of advantages compared to traditional methods for remote sensing of urban T_surf:

• T_{hem, r} samples the surface hemispherically (i.e., samples vertical, horizontal, and sloped features), providing a temperature that is more geometrically representative than one from a narrow field-of-view remote sensor in the nadir [10].
• T_{hem, r} is time-continuous and derived from measures that are often made for periods of a year or more. This allows for continuous analysis of urban T_surf at a wide range of time scales.

These advantages help to address question two posed in Roth et al. [1] by providing a more complete understanding of geometric and temporal biases in traditional methods for remote sensing of urban T_surf and by providing a more thorough description of the “true” geometric and temporal character of urban T_surf. Although T_{hem, r} provides a unique, time continuous, hemispherical perspective on urban T_surf it is subject to a number of disadvantages:

• The wide spectral response of a pyrgeometer makes it potentially more susceptible to atmospheric effects compared to radiometers that operate over smaller, more transparent, spectral ranges.
• T_{hem, r} is not spatially extensive. The method yields a single value that is representative of a view factor weighted average temperature of all of the surfaces “seen” by the sensor (analogous to a single pixel of a thermal image from a satellite remote sensor).
• When measured via a pyrgeometer, longwave radiation upwelling from the urban surface is spatially variant when measured from heights below approximately 3 to 5 times mean building height. For a pyrgeometer mounted below this threshold, T_{hem, r} is potentially biased towards surfaces that are closest to the sensor.

In spite of these disadvantages, this method can be used to supplement the existing urban T_surf record, to quantify its geometric and temporal biases, and to provide for analysis of urban T_surf over a wide range of time scales. To date, similar characterizations of the effect of complex surface geometry on remote sensed urban T_surf are restricted to short, often expensive ground [11] and aerial [12] transect campaigns and intensive observation periods [13,14], and this restricts analysis to sub-seasonal scales.

1.1. Describing Radiation as Received by a Remote Sensor

As radiation passes from an emitting surface to a remote sensor, it interacts with the atmosphere. This interaction can result in atmospheric effects on the remotely sensed signal, as some of the energy emitted from the surface may be absorbed or scattered. The atmosphere may also emit and scatter radiation towards the sensor, further confounding the remotely sensed signal. These effects vary spectrally and as a function of the distance between the surface and the sensor and the density of absorbing/scattering constituents in the layer. For a remote sensor “seeing” broadband longwave radiation, such as a pyrgeometer, atmospheric effects are largely a function of the absolute humidity and temperature of a layer and, as shown in Figure 1, are particularly variable at relatively short path lengths.

Radiation flux passing through a layer of atmosphere from a Lambertian surface towards a sensor can be described in a number of ways: Spectral directional radiance $R_z^{\uparrow }(\lambda ,\theta ,\varphi )$, at height z, wavelength $\lambda$ and from a direction defined by viewing zenith angle $\theta$, and azimuth angle $\varphi$ can be written as

$$R_z^{\uparrow }(\lambda ,\theta ,\varphi )={\tau }_{\lambda ,\theta ,\varphi }{ϵ}_{\lambda ,\theta ,\varphi }{R}_{0,\lambda }^{\uparrow }+(1-{ϵ}_{\lambda ,\theta ,\varphi })R_{sky,\lambda }^{\downarrow }+(1-{\tau }_{\lambda ,\theta ,\varphi })R_{atm,\lambda }^{\uparrow }$$

(1)

where $ϵ$ is spectral surface emissivity, $\tau$ is spectral “slab” transmittance for a given wavelength over the path length from $z=0$ to z. $\tau$ and $ϵ$ represent a single $\lambda$, $\theta$, and $\varphi$, while spectral directional radiances upwelling from the atmosphere $R_{atm}^{\uparrow }$, the surface $R_0^{\uparrow }$, and downwelling from the sky $R_{sky}^{\downarrow }$ depend on $\lambda$. $R_0^{\uparrow }$, $R_{atm}^{\uparrow }$, and $R_{sky}^{\uparrow }$ can be described spectrally as Planck’s law

$${\displaystyle R(\lambda ,\theta ,\varphi )=\frac{C_1}{\pi {\lambda }^5\left(exp\left(\frac{C_2}{\lambda {\mathrm{T}}_{\theta ,\varphi }}\right)\right),}}$$

(2)

where C₁ = $3.7404\times {10}^8$ $\mathrm{W} \mathsf{\mu }{\mathrm{m}}^{-4} {\mathrm{m}}^2$, C₂ = 14,387 $\mathsf{\mu }\mathrm{m} \mathrm{K}$, and ${\mathrm{T}}_{\theta ,\varphi }$ is emitter temperature.

Measured by a narrow field-of-view (FOV) sensor mounted at height z, $R_z^{\uparrow }(\lambda ,\theta ,\varphi )$ passes through an instrument filter (or dome) with spectral transmittance ${\tau }_d\left(\lambda \right)$ and is integrated over the sensor waveband (${\lambda }_1-{\lambda }_2$) to yield a directional radiance $L_z^{\prime }$ as “seen” by the sensor

$$L_z^{\prime }(\theta ,\varphi )={\int }_{{\lambda }_1}^{{\lambda }_2}{\tau }_d\left(\lambda \right)R_z^{\uparrow }(\lambda ,\theta ,\varphi )d\lambda$$

(3)

which, integrated over the hemisphere with respect to zenith angle $\theta$ and azimuth angle $\varphi$, yields an irradiance L at height z,

$$L_z={\int }_0^{2\pi }{\int }_0^{\pi /2}{L}_z^{\prime }(\theta ,\varphi )cos\theta sin\theta d\theta d\varphi .$$

(4)

1.2. Relating TIR and Surface Temperature

TIR radiation received by a remote sensor can be related to a surface temperature in a number of ways—each producing different conceptions of T_surf from different instrument and sensor types. As such, the term “surface temperature” with respect to a remote sensed TIR radiation is vague and can refer to several definitions of “surface” and “temperature”. Thus, proper terminology must be attached to land T_surf inferred from TIR radiation. Definitions and nomenclature conventions for multiple methods for T_surf retrieval are discussed at length in Norman and Becker [16].

Irradiance $L_z$ received by a pyrgeometer, can be used to infer a hemispherical brightness temperature T_{hem, b} through an inversion of the Stefan–Boltzmann law,

$${\mathrm{T}}_{\mathrm{hem},\mathrm{b}}=\sqrt[4]{\frac{L_z}{\sigma }},$$

(5)

where $\sigma$ is the Stefan–Boltzmann constant.

A directional brightness temperature T_bright $(\theta ,\varphi )$ from some viewing angle described by $\theta$ and $\varphi$ can be inferred from directional radiance via Equation (5) by replacing $L_z$ with $L_z^{\prime }$ multiplied by a coefficient. This method is commonly used to infer T_bright$(\theta ,\varphi )$ from infrared thermometers (IRT) operating over an atmospheric window—a narrow spectral range in which atmospheric effects are minimal and T_bright is a reasonably accurate approximation of T_surf. However, constants must be calibrated for the range of expected T_surf as the relationship between $L_z$ and $L_z^{\prime }$ is not perfectly linear with respect to emitter temperature.

Inversions of uncorrected $L_z$ or $L_z^{\prime }$ yield a temperature equal to that of a blackbody emitting the same amount of radiation as detected by the sensor. Since $L_z$ is unlikely to be equal to $L_0$ and, by extension, $L_z^{\prime }$ is unlikely to be equal to $L_0^{\prime }$, T_{hem, b} at $z=0$ and T_{hem, b} at z often show significant deviation. Hence, T_bright and T_{hem, b} are generally considered only a rough approximation of radiometric T_surf.

To retrieve a more accurate estimation of the ‘true’ T_surf, the same inversions can be applied to TIR measurements after correction for atmospheric effects (e.g., modification of the remote sensed TIR signal to represent the same signal at $z=0$ emitted from a homogeneous, isothermal, blackbody emitter) to yield a directional radiometric surface temperature T_rad from atmospheric corrected directional radiances and a hemispherical radiometric surface temperature T_{hem, r} from atmospherically corrected irradiances. T_rad and T_{hem, r} provide a better approximation of the ‘true’ T_surf by representing the temperature at which emitting surfaces are radiating, integrated over the sensor FOV.

1.3. Atmospheric Correction of Near-Ground TIR Radiation Measured in Urban Environments

Atmospheric correction of near-ground remote sensed TIR radiation is subject to a unique set of challenges compared to traditional satellite and aerial platforms. Wide-FOV remote sensors have complex, multiple line-of-sight (LOS) path length geometries—illustrated in Figure 2 for a downward facing pyrgeometer. Surface-sensor geometry varies significantly over the sensor FOV as some path lengths intersect with raised vertical, sloped, and horizontal features. This creates the potential for non-uniform atmospheric effects over the sensor FOV and necessitates a multi-LOS correction to retrieve accurate T_{hem, r}. In effect, with near-ground wide-FOV sensors, surface geometry is non-trivial and must be represented in atmospheric correction routines. In contrast, over a scene retrieved via satellite, spatial viability in surface geometry and LOS angle have a negligible effect on path length. Atmospheric correction routines for satellite retrieved TIR radiation, therefore, assume uniform or single-LOS geometry because the TIR signal passes through a relatively constant volume of atmosphere over the projected sensor FOV, regardless of surface geometry.

Several multi-LOS correction routines have been developed to remove atmospheric effects on near-ground remote sensed TIR radiation: Meier et al. [17] describes a correction method for oblique angled thermal imagery of urban terrain. In the method, spatially distributed LOS geometries were calculated by linking each image pixel to the corresponding three-dimensional coordinates of its viewpoint on a digital building model (DBM). Path lengths were then calculated as the distance between each pixel’s corresponding location on the DBM and the sensor represented as the vanishing point of a three-dimensional pyramidic projection from the sensor’s location in the DBM. A pixel-by-pixel correction was then applied to remove atmospheric effects and retrieve T_rad for each pixel at 30 $\min$ intervals, resulting in a brief, time-continuous climatology of urban T_rad. However, the method uses thermal images in conjunction with a DBM to calculate path length geometries for each pixel’s LOS—a technique not possible with a pyrgeometer, which returns a single integrated value over the sensor FOV. Moreover, the target instrument operates over a narrow waveband with relatively uniform spectral sensor response, reducing the magnitude and variance in atmospheric transmission over the sensor response curve. Thus, the method is not directly generalizable to correct TIR radiation measured via pyrgeometer.

Kotani and Sugita [18] describes a method for correction of wide-FOV (pyrgeometer) TIR irradiances over a homogeneous planar surface. In this method, path lengths were calculated for six sensor heights. Radiances were then modeled using the LOWTRAN [19] radiative transfer code initialized at $5^{\circ }$ intervals and integrated over the hemisphere to retrieve irradiances for a suite of T_surf, and ambient T_air and humidities. The resulting lookup table (LUT) of values is then used to correct $L_z$ and quantify atmospheric effects on remote sensed $L_z$ measured from several sensor heights.

The methods described in Kotani and Sugita [18] and Meier et al. [17] are limited to flat terrain and narrow-FOV thermal imagers respectively. However, by combining elements from the two existing methods we design a method for atmospheric correction of TIR upwelling from complex terrain measured via near-ground pyrgeometer.

2. Methods

The “rolling lookup-table” method described in this study uses a sensor view model in conjunction with a radiative transfer code to model hemispherical irradiances upwelling from a simplified isothermal three-dimensional representation of the urban surface. In summary, the method (depicted in Figure 3) uses vertical profiles of measured T_air and humidity to model at-sensor spectral radiances at 5${}^{\circ }$ angular increments over the sensor FOV for a predetermined range of possible T_{hem, r} at each time-step. Spectral directional radiances are convolved by a pyrgeometer dome transmittance curve, integrated over the sensor waveband, and weighted for their respective angular view factor. Weighted directional radiances are then integrated over the hemisphere and aggregated into a LUT of modeled irradiance—T_{hem, r} pairings for each time step, unique to the vertical profile of measured T_air and humidity. Finally, for each time step, measured irradiances are matched with the closest modeled irradiances in the LUT to return an atmospherically corrected radiometric hemispherical surface temperature. This process is repeated at 30 $\min$ intervals to yield a continuous climatology of urban T_{hem, r}. The following sections introduce the study area for which the method was developed and describe the sensor view model, radiative transfer, and post-processing steps of the correction method.

2.1. Study Area

As discussed in Section 1.3, atmospheric correction of longwave irradiances measured from downward-facing, near-ground, wide-FOV sensors must account for complex surface geometry. Thus, routines to retrieve atmospherically corrected urban T_{hem, r} from upwelling longwave irradiances are inherently site specific. However, it is important to note that although correction magnitudes described in this paper is not necessarily generalizable, the correction method described in this paper can readily be adapted to different study sites, sensor types, and unique surface geometries.

With methodological generalizability in mind, a “rolling lookup table” atmospheric correction method was developed to retrieve radiometric T_{hem, r} for a climatology of upwelling longwave irradiances measured from above the Sperrstrasse street canyon in Basel, Switzerland instrumented as a part of the Basel Urban Boundary Layer Experiment (BUBBLE) [13]. Site location, measured meteorological and radiation variables, and morphological characteristics are included in

Table 1. A description of morphological parameters and measured variables for the selected urban and rural sites from the Basel Urban Boundary Layer Experiment. Modified from Rotach et al. [13] to include only relevant parameters.

Site	Location	Morphological	Meteorological Variables	Radiation
	Height	Characteristics 1	[No of Levels] 2	[No of Levels] 3
Basel Sperrstrasse	47.57${}^{\circ }$ N	$z_H$ = 14.6 $\mathrm{m}$	Tair [7]	Lup [3]
Urban street canyon	7.60${}^{\circ }$ E	${\sigma }_H$ = 6.9 $\mathrm{m}$	H [7]	Ldown [5]
Local Climate Zone: 2	255 $\mathrm{m}$ a.s.l.	H/W = 1.0	WV [12]	Kup [2]
		${\lambda }_C$ = 1.92	WD [1]	Kup [2]
		${\lambda }_P$ = 0.54	P [1]
		$\alpha$ = 11.0%

¹ $z_H$: average building height, ${\sigma }_H$: standard deviation of building height, ${\lambda }_P$: plan aspect ratio, ${\lambda }_C$: complete aspect ratio, H/W: local canyon height to width ratio, $\alpha$: surface albedo. ² H: humidity, WV: wind velocity, WD: wind direction, P: pressure. ³ L_up: upwelling longwave radiation, L_down: downwelling longwave radiation, K_up: upwelling shortwave radiation, K_down: downwelling shortwave radiation.

. Morphological parameters for the were calculated for a 250 $\mathrm{m}$ circular area surrounding the study sites using the method described in Grimmond and Oke [20].

For the eight-month period between December 2001 and July 2002 during BUBBLE, a triangular lattice tower was installed within the Sperrstrasse street canyon. Its location was offset towards the southeast facing wall near the along-canyon center of the canyon. Instruments to observe a full suite of meteorological variables and fluxes of heat, mass, and momentum were mounted at various levels on the tower. Profiles of T_air and humidity were measured at seven heights extending from 2.5 $\mathrm{m}$ to 31.5 $\mathrm{m}$ above the canyon floor (with the highest observation level at approximately 2.17 times mean roof level). Upwelling and downwelling short/longwave fluxes were obtained from Kipp and Zonen radiometers mounted at the lowest and highest measurement levels, with an additional downward facing pyrgeometer mounted at roof level near the center of the street canyon. In addition, during a summertime intensive observation period (IOP) fron 10th June through 9th July an array of narrow-FOV IRTs was installed to sample representative individual facet surface temperatures (T_facet). A schematic of the locations of IRTs and pyrgeometers within the Sperrstrasse canyon is included in Figure 4.

The BUBBLE Sperrstrasse site was chosen here for three primary reasons: (1) the site provided a continuous dataset of radiation and meteorological variables over the course of nearly one year for a representative mid-latitude city. This allowed for examination of urban T_surf the surface urban heat island effect (sUHI), and atmospheric correction magnitudes over a wide range of representative mid-latitude conditions; (2) inclusion of T_facet over the IOP allows for investigation of the effect of sensor FOV and viewing direction on remote sensed urban T_surf for common methods of urban T_surf retrieval; and (3) the site has been used in multiple validation exercises for urban climate models [21,22].

Facet surface temperatures measured during the summertime IOP allow for direct comparison of T_{hem, r} to common remote sensed representations of the urban surface calculated from weighted averages of wall (T_wall), road (T_road), and roof (T_roof) temperatures. Plan and complete aspect ratios are used to derive weightings for nadir remote sensed (T_plan) and complete (T_comp) representations of urban surface temperature - described in

Table 2. Weights applied to individual wall, roof, and road surface temperature components in order to calculate complete and nadir temperatures for the Basel Sperrstrasse street canyon.

	Road	NNW	SSE	NNW	SSE
		Roof	Roof	Wall	Wall
Complete (Tcomp)	0.36	0.16	0.16	0.16	0.16
Nadir (Tplan)	0.46	0.27	0.27	0.00	0.00

. T_comp represents a complete urban T_surf, where facet temperatures are averaged based on their proportion of the complete urban surface area, while T_plan represents the Sperrstrasse site as viewed by a narrow-FOV remote sensor in the nadir. To facilitate comparison over the IOP, T_{hem, r} and T_plan were divided by T_comp and averaged at each time step over the IOP to yield normalized mean T_{hem, r} and T_plan at 30 $\min$ intervals. Through comparison of T_{hem, r} to T_plan and T_comp we investigate the effect of sensor-surface geometry on remote sensed urban T_surf and quantify directional biases in common urban T_surf measurements.

2.2. Modeling Path Lengths of Three-Dimensional Terrain

The sheer number of unique path length geometries inherent with wide-FOV radiometry of urban areas makes full three-dimensional radiative transfer simulation difficult and computationally intensive. In this method, instead of calculating upwelling radiances from each point on the surface to the sensor, radiances are calculated for azimuthally averaged path lengths that represent the average surface-sensor geometry for each solid angle “sector” of the sensor FOV. To visualize this, imagine binning all possible path lengths in three dimensions between two zenith angles (e.g., the solid angle sector bounded by $\theta$ = 90${}^{\circ }$ and $\theta$ = 75${}^{\circ }$) in Figure 2. In a given bin, path lengths running parallel to the canyon axis will be much longer, while those perpendicular to the canyon axis may intersect with walls or rooftops and be much shorter. The mean path length in each bin represents the average distance from the surface to the sensor for the “slice” of the sensor FOV. This simplification reduces the computational time needed to model each irradiance—T_{hem, r} pairing, as angular radiances can be computed as a function of zenith angle alone and subsequently weighted and integrated three-dimensionally over the hemisphere.

To calculate surface-sensor geometries, the Surface-Sensor-Sun Urban Model (SUM) [23] is initialized with a simplified, orthogonal three-dimensional DBM that represents surface geometry of the surrounding area. SUM uses a three-dimensional array to represent surface morphology (x, y, and z), with z representing height above the x, y plane. Information describing each cell is included in an additional dimension (in this case, distance from the cell to the sensor). After specifying sensor position and FOV, the model determines which cells have an unobstructed line of sight to the sensor and calculates distance from “seen” patches to the sensor. Path lengths are binned at 5${}^{\circ }$ increments of zenith angle and averaged to return an azimuthally-independent mean path length for each bin. In addition, while retrieving path length geometries, SUM calculates view factors for each solid angle sector, which are later used to weight angular radiances in the hemispherical integration post-processing steps.

In relatively homogeneous urban areas, simplification from three to two dimensions does not reduce accuracy because a pyrgeometer returns a single “bulk” measure of upwelling irradiance—one that is integrated over the sensor FOV. Thus, when atmospheric conditions and surface radiative properties are reasonably constant for a given solid angle sector, there is no functional difference between resolving radiative transfer for every possible path length with its unique T_air and humidity profile and resolving radiative transfer for a single azimuthally averaged atmospheric profile and path length geometry combination. In urban environments that are highly heterogeneous—for instance, in areas with significant heating, ventilation, and air conditioning (HVAC) exhaust or large variations in surface radiative properties—care should be taken when simplifying radiative transfer as emissivity, atmospheric transmittance, and/or T_air may change significantly over the FOV and may not be accurately represented by averaging in each solid angle sector.

2.3. Modeling Hemispherical Irradiances

With path length geometries calculated in SUM, irradiances are modeled for each time step using version 4.1 of the MODerate resolution atmospheric TRANsmission radiative transfer code (MODTRAN) [15]. At each time-step, a range of possible temperatures is specified (T_spec) based on T_{hem, b} calculated from the measured irradiance with

$${\mathrm{T}}_{\mathrm{spec},\min }={\mathrm{T}}_{\mathrm{hem},\mathrm{b}}-6\mathrm{K},$$

(6)

defining a minimum T_spec before iterating over

$${\displaystyle {\mathrm{T}}_{\mathrm{spec},\mathrm{i}}={\mathrm{T}}_{\mathrm{spec},\min }+\sum _{i=1}^{n}0.5n,1\le n\le 32}$$

(7)

to retrieve a lookup table (LUT) of T_spec.

After the LUT is defined, at-sensor spectral radiances for each path length/zenith angle are modeled at an average urban emissivity of 0.95 [24] over a waveband of 0–2300 ${\mathrm{cm}}^{-1}$ for each T_spec. Profiles of T_air and humidity are retrieved from 30 $\min$ averages of conditions observed at the Sperrstrasse site. Averages of 30 $\min$ are used over raw 5 $\min$ values in order to smooth the input T_air and humidities and to cut down on the number of model runs needed to retrieve a long term climatology. The method can be adapted to any time interval. Aerosol, trace gas concentrations, and above-sensor T_air and humidity conditions are defined by the mid-latitude summer standard atmosphere when daytime T_{air, max} > 10 ${}^{\circ }\mathrm{C}$ (the mid-latitude winter profile is substituted on days where T_{air, max} < 10 ${}^{\circ }\mathrm{C}$) [25].

In this method, a “typical” longwave bandpass—approximately 250–2300 $\mathrm{c}{\mathrm{m}}^{-1}$ (4–42 $\mathsf{\mu }\mathrm{m}$)—was extended to include much smaller wavenumber (longer wavelengths). This was done for two reasons: (1) to accurately represent broadband spectral longwave emission curves, which show significant emittance in wavenumber smaller than 250 ${\mathrm{cm}}^{-1}$ (wavelengths longer than 42 $\mathsf{\mu }\mathrm{m}$); and (2) to replicate the spectral signal “seen” by a silicone-domed pyrgeometer, which continues to transmit radiation at wavenumber < 250 ${\mathrm{cm}}^{-1}$ (wavelengths > 42 $\mathsf{\mu }\mathrm{m}$). A “typical” longwave bandpass underestimates a pyrgeometer signal by approximately 7–10 $\mathrm{W}{\mathrm{m}}^{-2}$, depending on emitter temperature. A comparison of ‘typical’ and ‘extended’ longwave bandpasses is shown in Figure 5.

To replicate the signal received by the sensor, at-sensor spectral radiances computed by MODTRAN are convolved by a dome transmittance curve and integrated over the bandpass via

$$L_z^{\prime }(\theta ,\varphi )=\frac{{\int }_{{\nu }_1}^{{\nu }_2}R(\theta ,\varphi ,\nu )r\left(\nu \right)d\nu }{\overline{r}}$$

(8)

to yield a directional radiance $L_z^{\prime }$ with units $\mathrm{W}{\mathrm{m}}^{-2}{\mathrm{sr}}^{-1}$ for each $\theta$ over a waveband of ${\nu }_1$ = 0 $\mathrm{c}{\mathrm{m}}^{-1}$ to ${\nu }_2$ = 2300 ${\mathrm{cm}}^{-1}$. $\overline{r}$ is Planck weighted mean broadband sensor response computed as

$$\overline{r}=\frac{\int R\left(\nu \right)r\left(\nu \right)d\nu }{\int R\left(\nu \right)d\nu },$$

(9)

where $R\left(\nu \right)$ is spectral radiance computed from a Planck function at an approximated emitter temperature and $r\left(\nu \right)$ is spectral sensor response.

$L_z^{\prime }\left(\theta \right)$ are then multiplied by their associated angular view factor weighting ($\mathsf{\Phi }$) and integrated over the hemisphere to yield an irradiance $L_z$ with units $\mathrm{W}{\mathrm{m}}^{-2}$ with for the target T_{hem, r}

$$L_z\left({\mathrm{T}}_{\mathrm{spec}}\right)={\int }_0^{2\pi }{\int }_0^{\pi /2}{L}_z^{\prime }(\varphi ,\theta )\mathsf{\Phi }\left(\theta \right)d\theta d\varphi .$$

(10)

$L_z\left({\mathrm{T}}_{\mathrm{spec}}\right)$ is representative of at-sensor irradiance upwelling from the urban surface described in SUM at the target specified temperature T_{hem, r} for the measured T_air, humidity, aerosol, and trace gas profile. The process is repeated to retrieve $L_z$ for the range of potential T_{hem, r} at the given time-step. Irradiance—T_{hem, r} pairings are then aggregated into a LUT. Finally, the measured irradiance is matched with its closest modeled irradiance to yield an atmospherically corrected, radiometric hemispherical surface temperature for the given time step. The workflow is repeated at 30 $\min$ intervals to yield a time series of T_{hem, r}. Atmospheric correction magnitudes can then be calculated as the difference between T_{hem, r} and T_{hem, b}, with T_{hem, b} calculated via Equation (5).

2.4. A Practical Parameterization

To save computational time and facilitate broader use of the method, a parameterization scheme was developed to simplify the correction process. In the parameterization, a suite of radiative transfer simulations is run prior to correction (as opposed to at each time step) to retrieve transmittances ($\tau$) for a range of humidities and path lengths. T_surf and T_air are held constant in these simulations as they have a relatively small effect on $\tau$. Simulations for view factor weighted path lengths from 1 m to 48 $\mathrm{m}$ and water vapor mass densities from 0.1 to 23.5 $\mathrm{g}/{\mathrm{m}}^{-3}$ are included in Appendix A of [26]. Sensor-surface geometry is approximated as a single view factor weighted, azimuthally averaged, surface-to-sensor path length calculated using the SUM [23] and a simplified DBM. This path length represents the average distance from the sensor to the surface as “seen” by the sensor. For each time step, measured humidity is used to infer a hemispherical transmittance ($\overline{{\tau }_{\mathsf{\Phi }}}$) for the approximated surface-sensor geometry. A parameterized T_{hem, r} can then be calculated by using measured (in this case, profile averaged) T_air to decompose measured upwelling longwave into irradiance received by the sensor from the surface $L_{surf}^{at-sensor}$ and from the atmosphere $L_{atm}^{at-sensor}$ via

$${\mathrm{T}}_{\mathrm{hem},\mathrm{r}}={\left({\displaystyle \frac{\left(\frac{{\displaystyle L_{surf}^{at-sensor}}}{{\displaystyle \tau }}\right)}{{\displaystyle \sigma }}}\right)}^{0.25},$$

(11)

where $\tau$ is calculated using a radiative transfer code and $L_{surf}^{at-sensor}$ is an estimation of the fraction of at sensor measured irradiance emitted by the surface calculated as

$${\displaystyle L_{surf}^{at-sensor}=L_{total}^{at-sensor}-(1-\tau )\sigma {{\mathrm{T}}_{\mathrm{air}}}^4.}$$

(12)

Using this parameterization, a climatology of T_{hem, r} for a given surface geometry can be retrieved from approximately 100 simulations as opposed to the many thousands needed for full hemispherical radiative transfer simulation. A variety of online and offline, open and closed source resources are available for radiative transfer simulation [15,27,28], some of which are available for free or at a low cost (e.g., Spectral-Calc, DART). Figure 6 and

Table 3. Statistical performance of T_hem derived using the parameterization scheme relative to modeled T_hem,r. MAE is mean absolute error. RMSE_s and RMSE_u represent the systemic and unsystematic root-mean-square error, respectively. Statistical tests were selected from [29,30]; n = 853.

Statistic	Parameterization
Slope	1.058
Intercept, ($\mathrm{K}$)	−1.488
R2	0.991
MAE, ($\mathrm{K}$)	0.607
RMSE, ($\mathrm{K}$)	0.757
RMSEs, ($\mathrm{K}$)	0.476
RMSEu, ($\mathrm{K}$)	0.588
d, agreement index	0.998

compare T_{hem, r} from the “rolling lookup-table” method and the parameterization. Errors are largest during the daytime hours, where the profile averaged T_air overestimates above canyon T_air and underestimates near-surface T_air. Neutral stability in the nighttime and early morning hours reduces these errors significantly as the canyon T_air profile is approximately isothermal.

3. Results

3.1. Method Evaluation

MODTRAN has been shown to effectively model near-ground radiative transfer at urban canyon scale path lengths in two dimensions [31,32]. However, the method described here accounts for complex three-dimensional surface geometry, which can exert strong influence on atmospheric transmittance. In addition, the method includes significant post-processing to weight and integrate point-to-point spectral directional radiances to derive three-dimensional, hemispherical irradiances. Thus, prior to deriving a climatology of urban T_{hem, r}, the method was evaluated using profiles of upwelling longwave irradiances measured over a simple flat surface. By testing the method using profiles of upwelling longwave radiation measured over a flat, relatively homogeneous, simple surface, we can assume that, although each sensor has a different set of weightings for each patch on the ground (i.e., the pyrgeometer closest to the ground has a larger view factor for the patch directly below the sensor than the highest pyrgeometer), each sensor should “see” approximately the same surface temperature. Therefore, any differences in the signal between sensor heights is solely a product of atmospheric influence—which should be represented by MODTRAN given concurrent profiles of T_air and humidity. Thus, it follows that, provided urban geometry and path lengths are accurately represented, the method should have similar accuracy over rough urban terrain.

For a continuous 14-day period, profiles of up/downwelling short/longwave radiation, T_air, and humidity measured from 2 $\mathrm{m}$, 10 $\mathrm{m}$, and 30 $\mathrm{m}$ were obtained at 30 $\min$ averages from an instrumented tower in Payerne, Switzerland. The tower, installed over a cultivated field as a part of the Baseline Surface Radiation Network (BSRN), is located approximately 100 $\mathrm{k}\mathrm{m}$ southwest of the BUBBLE Sperrstrasse tower and is subject to similar summertime conditions as the study site. The study period was chosen to include a comprehensive range of T_surf, T_air, humidities, and cloud coverages to represent typical summertime conditions over which atmospheric effects on TIR radiation can vary significantly. To evaluate the method, we used a modified version of the workflow described in Figure 3. First, azimuthally averaged path lengths over a flat surface were calculated in SUM at 5${}^{\circ }$ increments of zenith angle over the sensor FOV for the 10 $\mathrm{m}$ and 30 $\mathrm{m}$ sensors. For each time step, T_{hem, b} calculated from upwelling longwave measured at 2 $\mathrm{m}$ was used to model irradiances at 10 $\mathrm{m}$ and 30 $\mathrm{m}$ at 30 $\min$ intervals using concurrent profiles of T_air and humidity. A daytime warm bias of approximately 2 $\mathrm{K}$ was observed in the 30 $\mathrm{m}$ T_air measurement, even under high wind velocities when one would expect little vertical difference in temperature and certainly not a strong temperature inversion. The fact that the observed warming bias remains relatively constant in the daytime hours over the 14-day study period prompted a slight modification of the T_air profile to avoid linear interpolation between the 10 $\mathrm{m}$ T_air and the anomalously warm 30 $\mathrm{m}$ T_air. Additional T_air measurements were interpolated at 5 $\mathrm{m}$ increments starting from the 10 $\mathrm{m}$ measurement, using a lapse rate informed by the difference between T_air at 2 $\mathrm{m}$ and 10 $\mathrm{m}$ to replicate typical summertime lapse rates above flat vegetated terrain [24]. When a large warming bias was observed at 30 $\mathrm{m}$ in the hours after midmorning (when any inversion is likely to have dissipated), the 30 $\mathrm{m}$ measurement was modified to replicate neutral conditions in the hours around noon, and weakly unstable conditions in the afternoon and evening hours. These modifications provide a smooth T_air profile as one would expect over flat terrain with mild to moderate wind velocities. The nighttime profile was not altered. These modifications are expected to better represent the actual T_air profile in the layer between 10 $\mathrm{m}$ and 30 $\mathrm{m}$ than a simple linear interpolation over the layer.

Contrary to the findings in Kotani and Sugita [18], an isothermal atmospheric profile was not sufficient to accurately model upwelling fluxes over flat terrain. Although thermal stratification is relatively small by day in an urban canyon [33]—where strong microscale contrasts in T_surf foster strong canyon mixing and neutral stability—the large daytime T_surf − T_air differential and the path length/transmittance gradient can create large differences in upwelling longwave forced by inter-canyon thermal contrasts—a phenomenon termed radiative divergence. By underestimating near-surface T_air and overestimating canopy layer T_air, daytime divergences are underestimated by an isothermal profile. As such, a full canyon T_air and humidity profile is preferred to most accurately model near ground fluxes.

Modeled and measured upwelling longwave at 10 $\mathrm{m}$ and 30 $\mathrm{m}$ fluxes and divergences—calculated as the difference between 30 $\mathrm{m}$ and 10 $\mathrm{m}$ fluxes—are compared in Figure 7 and

Table 4. Statistical performance of the Payerne evaluation calculated at 30 $\min$ intervals over a continuous 14-day period. MAE is mean absolute error. RMSE_s and RMSE_u represent the systemic and unsystematic root-mean-square error respectively. Statistical tests were selected from [29,30]; n = 378.

Statistic	10 m Flux	30 m Flux	Divergence
Slope	1.002	1.011	0.792
Intercept, ($\mathrm{W}/{\mathrm{m}}^2$)	0.422	−3.560	−0.583
R2	0.995	0.996	0.863
MAE, ($\mathrm{W}/{\mathrm{m}}^2$)	1.781	1.325	1.191
RMSE, ($\mathrm{W}/{\mathrm{m}}^2$)	2.204	1.690	1.423
RMSEs, ($\mathrm{W}/{\mathrm{m}}^2$)	1.444	0.851	1.028
RMSEu, ($\mathrm{W}/{\mathrm{m}}^2$)	1.666	1.460	0.985
d, agreement index	0.999	0.999	0.959

. Both fluxes and divergences show strong correlation. Therefore, we conclude: first, irradiances measured near screen level (approximately 2 $\mathrm{m}$ above ground level) are not subject to significant atmospheric influence—as T_{hem, b} inferred from $L_{z=2\mathrm{m}}$ was sufficient for modeling irradiances at 10 $\mathrm{m}$ and 30 $\mathrm{m}$. As such, T_{hem, b} and T_{hem, r} are reasonably equal when z = 2 $\mathrm{m}$. However, it should be noted that, in urban areas, a 2 $\mathrm{m}$ sensor is not sufficiently representative of canyon geometry and should not be used to derive urban T_{hem, r}. Second, MODTRAN can accurately model longwave fluxes and divergences above a flat, homogeneous surface over a wide range of temperatures, humidities, and cloud coverages and large contrasts in near-ground stability. At 30 $\mathrm{m}$ under typical summertime profiles of humidity, CO₂, and O₃, view factor weighted hemispherical transmittance over a flat surface is approximately 55%—meaning 45% of a broadband hemispherical TIR signal at 30 $\mathrm{m}$ is emitted by the atmosphere, rather than the target surface. As such, it is imperative that T_air profiles are well represented in MODTRAN to accurately model the atmospheric component of remote sensed TIR signal. As profiles of T_air in the layer between the surface and the sensor at the Sperrstrasse site are generally subject to neutral or weakly unstable conditions, with only rare stable stratifications, urban longwave divergences are likely to be smaller than those observed in this evaluation—and well approximated by MODTRAN. Thus, provided path length geometries are accurately replicated by the sensor view model, we can reasonably assume that the method will have similar effectiveness at the urban site. Direct validation of the method in urban environments is difficult as the geometry and heterogeneous surface types of cities (and their associated micro-scale contrasts in T_surf) are difficult to accurately replicate in radiative transfer models.

3.2. Atmospheric Correction Magnitudes

Atmospheric correction magnitudes binned for season and time of day for the eight-month study period are shown in Figure 8. Summertime correction magnitudes show large day-to-day and diurnal variations. Variation is largest in summer near solar noon, where different synoptic conditions can force correction magnitudes of up to 8 $\mathrm{K}$ in clear-sky hot conditions and below 0 $\mathrm{K}$ under cooler overcast conditions. In winter, correction magnitudes display less variance and are smaller—approximately 2 $\mathrm{K}$ over a range of representative wintertime synoptic conditions.

Over both summer and winter seasons, correction magnitudes are strongly correlated with $\Delta$T_{hem, r−air}. Figure 9 shows the effect of seasonality and cloud cover on the relationship between correction magnitudes and $\Delta$T_{hem, r−air}. The relationship is strongest and most evident in summer when variations in cloud cover can lead to large day-to-day contrasts in solar input and $\Delta$T_surf−air, which translates into a wide range of correction magnitudes.

Figure 10 illustrates the relationship between correction magnitude and meteorological variables commonly measured as a part of climatological assessments of urban environments. A strong positive relationship is observed between correction magnitudes and incoming solar radiation. T_{hem, r} and T_air display more complex relationships with correction magnitude, with a large cluster of observed correction magnitudes situated at a local minimum between 10 ${}^{\circ }\mathrm{C}$ and 20 ${}^{\circ }\mathrm{C}$, weakly negative correlation at temperatures lower than 10 ${}^{\circ }\mathrm{C}$, and weakly positive correlation at temperatures greater than 20 ${}^{\circ }\mathrm{C}$. Atmospheric water vapor content did not appear to exert significant control over correction magnitudes.

Figure 11 shows hemispherical atmospheric transmittance as a function of vapor pressure for pyrgeometers at 10 $\mathrm{m}$ and 30 $\mathrm{m}$ above a hypothetical flat homogeneous surface. Transmittances are modeled in MODTRAN 4.1 as fraction of surface emission reaching the sensor. Hemispherical transmittance falls sharply with increasing vapor pressure between 0 $\mathrm{h}\mathrm{Pa}$ and 5 $\mathrm{h}\mathrm{Pa}$, but is relatively constant with vapor pressure >5 $\mathrm{h}\mathrm{Pa}$. This results in the weak correlation observed in Figure 10 between humidity and correction magnitude.

3.3. Comparing T_surf from Different Sensor Geometries

To illustrate the effect of sensor-surface geometry on remote sensed T_surf, Figure 12 shows a comparison of mean normalized T_{hem, r} and T_plan calculated by dividing T_{hem, r} or T_plan by T_comp at each timestep and averaging over a 14-day subset of the IOP from 26 June 2002 through 9 July 2002 during which all IRTs were operational. In the figure, overestimation of T_comp is observed at values above one and underestimation by values below one. Both T_{hem, r} and T_plan overestimate T_comp by day and underestimate T_comp by night. Mean daytime overestimation of T_comp by T_{hem, r} is smaller and less variable than that by T_plan.

Mean normalized T_{hem, r}, T_plan, and $\Delta$T_{plan−hem, r} are shown in Figure 13 along with clear sky and overcast sky case days. While mean T_{hem, r} and T_plan display similar overestimations of T_comp, under clear sky conditions, overestimation by T_plan is much larger than that by T_{hem, r}.

4. Discussion

4.1. Controls on Atmospheric Correction Magnitude

Atmospheric correction magnitudes of downward facing pyrgeometers are highly dependent on factors that control $\Delta$T_surf−air. With a non-zero $\Delta$T_surf−air, TIR radiation emitted from the surface and absorbed by the atmosphere is re-emitted at a different temperature, modifying the at-sensor TIR signal. Thus, the strength of the T_surf to T_air differential is directly related to the magnitude of atmospheric effects on a remote sensed TIR signal. This results in a strong positive relationship between atmospheric correction magnitude and $\Delta$T_{hem, r−air}. With a non-zero $\Delta$T_surf−air, TIR radiation emitted from the surface and absorbed by the atmosphere is re-emitted at a different temperature, modifyingair. Results in Figure 9 and Figure 10 show that conditions that maximize microscale surface to atmosphere thermal contrasts result in large correction magnitudes and conditions that suppress differences between T_surf and T_air show much smaller or negative correction magnitudes.

Of the meteorological variables tested, correction magnitudes are most strongly dependent on solar input. Solar radiation incident on the surface forces microscale thermal contrasts between the surface and the air, directly controlling the T_surf − T_air difference. The relationships between correction magnitude and T_{hem, r} and T_air individually are weaker and more complex. Correction magnitudes are smallest when T_{hem, r} and T_air are between approximately 10 ${}^{\circ }\mathrm{C}$ and 20 ${}^{\circ }\mathrm{C}$—largely made up of morning and cloudy observations during the summer months where thermal contrasts between T_surf and T_air are small or weakly negative, forcing small correction magnitudes. During the winter months, a negative relationship between T_{hem, r} and T_air and correction magnitudes is observed, likely a result of the effect of shifting spectral emittance curves combined with non-uniform sensor response—discussed in Section 4.2. On clear sky summer days, positive relationships between correction magnitudes and T_{hem, r} and T_air are observed as an increased solar input foster higher overall temperatures and large contrasts between T_surf and T_air.

Results in Figure 8 show significant seasonal and diurnal variation in correction magnitudes. The range of observed correction magnitudes is largest in summer, when a wide range of mid latitude synoptic conditions result in significant contrasts in daytime solar input and consequently large variations in the T_surf − T_air difference. Maximum correction magnitudes over the study period (often approaching 7 $\mathrm{K}$ to 8 $\mathrm{K}$) occur frequently near solar noon on clear, hot, humid days, during which intense solar heating of the surface produces a large T_surf − T_air differential. Minimum correction magnitudes near $-1$ $\mathrm{K}$ occur consistently on clear, calm nights following hot days, when T_surf can dip below T_air, even in urban areas where canyon trapping of radiation reduces cooling rates for both T_surf and T_air. In winter, decreased variability in overall solar input results in less intense solar heating of the surface and smaller T_surf − T_air contrasts. Thus, winter correction magnitudes are smaller and more consistent with a mean of approximately 2 $\mathrm{K}$.

4.2. The Effect of Non-Uniform Pyrgeometer Spectral Dome Transmittance on Correction Magnitudes

A pyrgeometer’s dome transmittance is spectrally non-uniform. Thus, the “dome effect” a pyrgeometer exerts on an incoming TIR signal changes as a function of emission temperature (T_emiss) because the shape of a given spectral radiance curve (R_$\nu$) is temperature dependent. This is illustrated in Figure 14 by comparing the relative peaks and shape of a pyrgeometer dome transmittance curve for several R_$\nu$ (T_emiss) for a range of typical urban T_emiss. Each curve interacts with a different portion of the spectral dome transmittance curve (r_$\nu$) resulting in different effects from the pyrgeometer dome for each T_emiss.

When modeling a pyrgeometer signal, it is important to account for the influence of a specific pyrgeometer dome’s spectral transmittance to avoid introducing a temperature dependent bias and to ensure the modeled signal is accurate. As such, the method in this study convolves modeled at-sensor spectral radiances by a spectral dome transmittance function obtained from Kipp and Zonen to accurately model upwelling irradiances as “seen” by the pyrgeometer.

To understand the influence of differential pyrgeometer dome transmittance effects at typical urban T_emiss, we use Equation (9) to calculate Planck weighted dome transmittances $\overline{r}$ at 5${}^{\circ }$ intervals from T_emiss = 260 $\mathrm{K}$ to T_emiss = 320 $\mathrm{K}$, shown in

Table 5. Planck weighted mean spectral dome transmittance $\overline{r}$ for a suite of common T_surf.

Temiss, $\mathbf{K}$	$\overline{\mathit{r}}$, %	Temiss, $\mathbf{K}$	$\overline{\mathit{r}}$, %
260	48.3	295	49.6
265	48.5	300	49.7
270	48.7	305	50.0
275	48.9	310	50.1
280	49.1	315	50.3
285	49.3	320	50.4
290	49.4

. $\overline{r}$ varies by approximately 2% over typical urban T_emiss, with higher $\overline{r}$ at low T_emiss and lower $\overline{r}$ at high T_emiss. Although this bias appears small, when comparing observations from multiple pyrgeometer types (each with their own spectral dome transmittance function) and when comparing modeled and measured irradiances, these biases may contribute an additional source of error.

4.3. The Effect of Sensor Sampling Geometry on Remote Sensed T_surf

Results in Figure 12 and Figure 13 show that sensor sampling regimes can have a significant effect on remote sensed T_surf. Both T_{hem, r} and T_plan overestimate T_comp by day and underestimate T_comp by night. These results are consistent with findings in [10,34] observed for clear sky days. However, when analyzed over a time series, over/underestimations from geometric effects are highly dependent on synoptic conditions, particularly for a nadir view of the urban surface. T_plan is much greater than T_comp under clear sky conditions, with T_{hem, r} showing a smaller positive bias. Over half of the T_plan signal from the Sperrstrasse canyon is generated by rooftops that have both a large diurnal T_surf amplitude and the highest daytime facet T_max. Undersampling of sloped facets and neglect of wall facets by T_plan also has a significant effect on the daytime clear-sky warming bias, as walls show much cooler daytime T_max than T_roof or T_road and have a moderating effect on T_comp and T_{hem, r} by day, particularly on hot clear sky days. Mean daytime overestimations by T_plan just after solar noon are approximately 4 $\mathrm{K}$ during the IOP, approaching 8 $\mathrm{K}$ on the clear sky case day. T_{hem, r} shows smaller overestimations of approximately 3 $\mathrm{K}$ over the 14-day subset of the BUBBLE IOP and 5 $\mathrm{K}$ on the clear sky case day. Urban T_surf and the sUHI are often measured from satellite remote sensors that retrieve T_plan exclusively under clear-sky conditions and often sample in the nadir [35,36]. Thus, overestimation of daytime urban T_comp inherent in the satellite T_surf record is best represented by T_plan on the clear sky case day.

Nighttime underestimations of T_comp by nadir and hemispherical views are a result of an undersampling of wall and road facets, respectively, as suggested by Roth et al. [1]. T_{hem, r} derived from $L_z$ measured at the Sperrstrasse canyon oversamples the nearest roof and underestimates the canyon floor compared to T_comp as its location along the canyon axis is skewed towards the south facing wall (rather than the center of the canyon) and its height is approximately 2.17 times mean building height. Sub-optimal sensor placement in the Sperrstrasse canyon—discussed in a sensor placement sensitivity test included in Appendix A—results in a large underestimation of nighttime T_comp by T_{hem, r}. However, nighttime underestimation—and to some extent, daytime overestimation—by T_{hem, r} is the combined result of a sampling bias inherent in a hemispherical view of the surface and improper sensor placement. Both of these biases can likely be reduced significantly by adjusting sensor placement to best represent surface geometry.

5. Conclusions

Accurate, geometrically representative, spatially extensive, and temporally continuous thermal remote sensing of complex terrain (urban or otherwise) is difficult. It is unlikely that a single sensor will satisfy spatial, geometric, and temporal goals. Thus, a combination of satellite, aerial, and in situ observational methods has been integral in isolating and elucidating the urban effect on surface temperature and its spatial, temporal, and geometric characteristics. However, neither technological advancement nor critical analysis of these methods has provided a satisfactory answer to the questions posed in Roth et al. [1] viz. urban thermal remote sensing. As such, the true spatial, temporal, and geometric nature of the urban effect on surface temperature remains difficult to measure and quantify.

This paper details a method for urban surface temperature (T_surf) retrieval using atmospherically corrected hemispherical radiometric surface temperatures derived from continuous near-ground measurements of thermal infrared radiation. Atmospherically corrected, hemispherical urban T_surf is derived using a method that combines a sensor view model (SUM) to represent urban surface geometry and a point-to-point radiative transfer code (MODTRAN 4.1) to model irradiances upwelling from complex surface terrain in three-dimensions. At each time step, irradiances are modeled using profiles of air temperature (T_air) and water vapor content for a range of possible hemispherical radiometric T_surf (T_{hem, r}). Irradiance–T_{hem, r} pairings are aggregated into a lookup table and the measured irradiance is matched with the closest modeled irradiance to retrieve T_{hem, r} for a given time step. Repeated at 30 $\min$ intervals, the method is used to derive an eight-month climatology of urban T_{hem, r} from urban irradiances measured as a part of the Basel Urban Boundary Layer Experiment (BUBBLE) in Basel, Switzerland.

Atmospheric corrections are large and show large variations based on sensor–surface–sun geometry and synoptic conditions. Thus, T_hem derived from irradiances measured at different heights or above different surface geometries may have different atmospheric or dome effects that can confound results. This fact makes clear the need for robust atmospheric correction of T_hem for surface urban heat island (sUHI) analysis or to retrieve radiometric T_hem. Correction magnitudes are largest (approaching to 8 $\mathrm{K}$) on hot, clear sky summertime days—with large solar input and high T_surf and T_air—and smallest (1 to −2 $\mathrm{K}$) by night and under overcast conditions, during which solar input is suppressed and T_surf and T_air are low. Correction magnitudes are most strongly correlated with the T_surf to T_air differential and incoming solar radiation and weakly correlated with T_{hem, r} and T_air. Water vapor content did not appear to exert strong control over correction magnitudes.

Comparison of T_surf from nadir (T_plan), hemispherical, and complete (T_comp) representations of the Basel Sperrstrasse street canyon show that a hemispherical view is more geometrically representative of the complete surface temperature than a sensor viewing in the nadir. However, sensor placement sensitivity tests show that T_{hem, r} varies based on sensor placement and height. Overestimations of T_comp by T_plan and T_{hem, r} are greatest under clear sky conditions (up to 8 $\mathrm{K}$ for T_plan and 5 $\mathrm{K}$ for T_hem). This is particularly important when one considers that satellite observations of urban T_surf are only possible under clear sky conditions (when T_plan is least representative of T_comp). Significant variability in over/underestimation of T_comp by T_{hem, r} and T_plan based on time of day and synoptic conditions make these biases difficult to generalize to other urban study sites with different surface geometries and characteristics, canyon orientations, and macro climates. However, the measures used to derive T_{hem, r} are common to most energy balance assessments and constitute an untapped, but promising, resource to quantify the geometric and temporal biases inherent in satellite remote sensing of TIR radiation from complex terrain.

By providing for time-continuous measurement of urban T_surf, the method proposed here can be used to provide all weather, long-term seasonal and diurnal analysis of the surface urban heat island effect as demonstrated in [26]. However, it is important to note that use of the method for sUHI analysis requires a matched pair of urban and rural instruments, with the urban measurement taken from a tower three to five times mean building height. This limits use of the method to select local climate zones [37] as towers of optimal height are not feasible in areas with very tall buildings. As a result, the method will not necessarily diagnose the spatial maximum or minimum sUHI. Further care should be taken to accurately represent surface-sensor geometries, the spectral dome characteristics of the pyrgeometer, and the near-ground air temperature and humidity profile in radiative transfer simulations to ensure the correction for atmospheric effects is robust.