Accuracy of Mean Radiant Temperature Derived from Active and Passive Radiometry

: The concept of the mean radiant temperature ( T mrt ) allows the study of radiative exchanges between a human and its environment. It presupposes that the radiant e ﬀ ects on the person of the actual environment, which is generally heterogeneous, and the virtual environment, which is deﬁned as homogeneous, are identical. ISO 7726 speciﬁes the required accuracy in T mrt as input of rational thermal indices, outdoors ± 5 (K). T mrt accounts for the radiant heat absorbed by skin / clothing from the shortwave (SW) and longwave (LW) spectral bands. Most of the radiant components are isotropic. However, there are anisotropic SW components; namely the direct irradiance and under clear or partly obstructed skies a signiﬁcant circumsolar fraction ( f cs ) in the di ﬀ use irradiance. Both originate from the close proximity of the solar disk. This study highlights the e ﬀ ect of f cs on T mrt . In the scope of human biometeorology a standing body posture is standard. For unidirectional irradiances its radiant cross-section varies dependent on the solar altitude. Active radiometry in deriving T mrt is based on measured irradiances. One method is the Klima-Michel-Modell (KMM) that uses readily available measurements from standard meteorologically radiant observations. KMM references Fanger’s area projection factors that are derived from precise measurements of real humans. Thus, KMM serves as reference in evaluation of further methods. One is the six-directional instrument ( T mrt,r,6 − Dir ). Slightly simplifying a standing human, it represents a subject as a rectangular solid. T mrt,r,6 − Dir is derived based on measured irradiances incident on the vertical and horizontal planes. In passive radiometry the energy balance equation of a black globe thermometer is solved that leads to T mrt,Tg,BG . f cs signiﬁcantly impacts T mrt with noticeably reduced values for high and increased for low solar altitudes. Hence, accounting for f cs is essential for the accuracy of T mrt . For KMM an extension to an existing algorithm is provided in order to include f cs into the T mrt calculation that results in T mrt,r,KMM . For T mrt,r,6 − Dir the radiant cross-section of the solid depends to a minor extent on its azimuth relative to the solar azimuth. As a result T mrt,r,6 − Dir slightly scatters compared to T mrt,r,KMM . However, it remains within ± 2 (K). T mrt,Tg,BG compared to T mrt,r,KMM complies only at night with the ISO 7726 bin of ± 5 K. T mrt,Tg,BG signiﬁcantly overestimates T mrt,r,KMM during the daytime, because of its greater SW absorptance compared to skin / clothing and to a smaller extent because the standing posture is represented by a sphere. Particularly in sunny conditions, T mrt,Tg,BG is subject to considerable variance. Thus, outdoors during the daytime, T mrt,Tg,BG is unable to serve as an appropriate input for the calculation of rational-based thermal indices. down- and upward radiant ﬂux densities; direct and di ﬀ use solar irradiance components; anisotropic circumsolar fraction in di ﬀ use irradiance


Introduction
Human biometeorology is the interdisciplinary field of science that studies interactions between the human body and the Earth's atmosphere and it is thus related to outdoor environments. Especially, it is concerned with contingency planning, health protection, regional and urban planning, the design of open spaces, various aspects of tourism and recreation areas, as well as research in climate change [1]. Its results are used in heat warnings and to provide medical-related information for weather forecasting. All these facets require a rational, thermophysiologically consistent assessment of the human thermal environment. This is frequently accomplished by applying thermal indices. Rational thermal indices are based on complete human energy budget models consisting of both a controlled passive and a controlling active system [2]. They require six basic parameters in order to define the human thermal environment. These are four meteorological variables, namely air temperature (t a ( • C) or T a (K)), water vapor pressure (e a (hPa)), air velocity (v a (m/s)) and mean radiant temperature (T mrt (K) or t mrt ( • C)), plus two inputs that describe properties affecting the human body: the insulation of clothing worn and the activity of the person described in terms of net metabolic heat production [3].
Compared to t a , e a and v a , which can readily and accurately be measured, T mrt is the most complex of the four meteorological inputs [1,4]: it is the uniform temperature of an imaginary black enclosure (emissivity ε = 1), in which a human body of defined azimuth and posture would gain the same radiant heat as in the actual nonuniform enclosure [5][6][7][8]. T mrt assesses the impact of the solar shortwave (SW, 0.3-3 µm) and the terrestrial (thermal) longwave (LW, 3-50 µm) radiant components on the human energy balance.
The concept of T mrt allows the study of radiative exchanges between a human person and its environment. Because it presupposes that the radiant effects on the person of the actual environment which is generally heterogeneous and the virtual environment which is defined as homogeneous are identical, it facilitates thermophysiological modeling [5,8]. T mrt is a direct measure of the environmental sensible heat stress onto the body that come from radiation. The rational indices resulting from thermophysiological modeling and their application require sufficient accuracy of T mrt . Those are quantified by ISO 7726 according to the extent of the thermal annoyance to be assessed.
The calculation of T mrt is originally developed for indoor applications. In a room, T mrt may be calculated from the surface temperatures T N of the surrounding N surfaces with uniform temperature each by accounting for the angle factor F p−N between a person and the surrounding surfaces [6,8,9]: F p−N is a function of the shape, the size and the relative position of surface N in relation to the person. The position of the surface N is described in spherical coordinates by azimuth and elevation angles. The angle factor can precisely but elaborately be determined by numerical integration over the azimuth and elevation angles. The integration takes place between the normal of the person's location on the concerned surface up to the angle of N. F p−N consists of the projected area factor f p that is weighted according to the relative position of surface N to the person. f p = A p /A eff relates the projected area of a person A p to its effective radiant area A eff , i.e., the total surface of the person minus that area parts that overlap, e.g., arms and trunk. Thus, f p is a function of the body posture and the elevation angle [6,10,11] and is precisely determined by the shadow area of the irradiated person on a plane normal to the direction of the incident beam. As the sum of the angle factors is unity, the fourth power of T mrt will be seen to be equal to the mean value of the surrounding surface temperatures to the fourth power, weighted with the respective angle factors [6].
By inversion of the Stefan-Boltzmann law, the T 4 N in Equation (1) can be replaced by measured irradiances, i.e., by active radiometry. Thus, measured radiant flux densities of the effective components in the SW and LW spectral bands can directly be applied. This is the base for outdoor use within the scope of human biometeorology where surface temperatures of the upper (atmospheric) half space are in principle not available.
Atmosphere 2020, 11, 805 3 of 22 In applied human biometeorology an upright reference subject is used that is rotationally symmetric to its vertical axis because the azimuthal orientation of the subject is usually unknown.
Its elevation dependent f p is derivable from diagrams and tables [1,6,11]. Because only the fraction of radiant energy that is absorbed by the skin/clothing surface can have an effect on the human energy balance, the clothing/skin of the reference subject is assumed to show overall SW absorption (α sw,Sk ) of 0.70 and LW emissivity (ε lw,Sk ) of 0.97 that is identical with its absorptivity (Kirchhoff law) [1,8,9].
In the scope of human biometeorology, two methods that apply active radiometry are in use. One [12] bases on operationally available radiant flux densities as they, for example, are recorded by meteorological services according to guidelines of the World Meteorological Organization [13,14]. The other applies the definition of plane radiant temperature [8,9]. It uses measured SW and LW radiant flux densities onto the six planes of a rectangular solid that represents a human of standing body posture [1,15].
Another approach to determine T mrt is passive radiometry. Here, T mrt is indirectly derived by determining the energy balance of a measuring system, especially the black globe thermometer [8,16,17]. The method has a long history and provides in those early years at least estimates of T mrt where the measurement of radiant flux densities was complex due to technical reasons and expansive [16,17]. The black globe was originally introduced in order to measure radiant heat indoors in relation to human comfort [17]. However, it is also applied outdoors, where it is integrated in instruments that provide the measured base for the calculation of the wet-bulb-globe-temperature index [18,19]. In the scope of human bioclimatology, the black globe is now also being used outdoors in the form of a matt black-painted table-tennis ball [20,21]. T mrt derived from the black globe is an approximation due to the difference in shape between a person and a globe [8]. In particular, direct SW irradiation at high solar altitudes will be over-estimated by the globe in relation to that received by a standing person. The black globe differs strongly in the SW absorptance compared to the outer surface of a clothed person. Furthermore, the accuracy of measuring T mrt using a black globe can vary to a great extent according to the values for the other characteristics of the environment [8,22].
The occurrence of heat stress situations is usually related to clear sky conditions, when humans are directly exposed to intense, frequently asymmetric irradiation. In this case, the diffuse radiant intensity from the sun's direction is substantially higher than from the remaining upper half space [23][24][25] and may significantly affect T mrt especially for the standing body posture.
This study aims to assess the three methods for outdoor use. In particular, it shall be evaluated whether the achieved accuracy is sufficient that the derived T mrt can serve as input for the calculation of thermal indices. Furthermore, it shall be investigated how the circumsolar anisotropy in the diffuse radiant component affects T mrt .
The objectives are accomplished by defining the effective solar and terrestrial irradiances in section "Methods and Materials", in which the methods and their theoretical background are shortly described and the available data for their evaluation are presented. The assessment methodology including the method selected for reference is introduced. Based on these data, the accuracy of the remaining methods is evaluated against the reference in section "Results" and the issue is examined how the circumsolar anisotropy in the diffuse SW irradiance affects T mrt .

Effective Solar and Terrestrial Irradiances
Solar electromagnetic irradiation that reaches the earth's surface covers a spectral range betweeñ 280 and 3000 nm. There are different impacts on the human organism depending upon the spectrum. The wavelengths between 400 and 730 nm enable photopic and scotopic vision. Especially in the UV-B range ≤ 315 nm, the impact on the skin may be harmful. Dependent on the dose and the skin type, it can provoke an erythema. However, the predominant impact of SW and LW irradiance on the skin is thermal. It is spectrally independent as long as the intensities in a given spectral range are less than 1000 (W/m 2 ). Thus, it is the total intensity or energy that affects a person's thermal psycho-physiological state [26]. Hence, it is to be concluded that T mrt can be derived for the relevant wavelength bands from spectrally integrating instruments. They are deployed as standard in meteorology.
The effective SW radiant flux densities (W/m 2 ), are • The downward direct irradiance (E I ) incident on a plane normal to the radiation's direction and measured by a pyrheliometer. E I is that part of the solar radiation that is not scattered by atmospheric constituents in its transfer to the earth's surface. Thus, E I is unidirectional because it originates from the solar disk, i.e., it is anisotropic.

•
The downward diffuse (directionally indistinguishable) irradiance (E D ) incident on a horizontal plane facing the upper half space and measured by a pyranometer combined with e.g., a shadow disk moved by a sun tracker or a shadow band. The latter requires subsequent correction of the reading, which accounts in particular for sky radiance anisotropy [23,27,28]. E D is that part of the solar radiation that has been subjected to scattering by atmospheric atoms and molecules (Rayleigh) and by aerosol particles and cloud droplets (Mie). Scattering changes the direction of the incident beam, yet leaves its energy and frequency unchanged. E D is multidirectional; frequently it impacts with the same amount of energy from all directions of the sky dome, and is then termed isotropic.

•
The surface reflected irradiance (E R ) above the lower half space and incident on a horizontal plane facing downward and measured by a pyranometer. E R is predominantly Lambertian equivalent reflected, i.e., it is diffuse. The direct mirror-like reflected proportion of E R remains small and can be regarded as insignificant [29,30]. Thus, E R is usually isotropic with regard to the lower half space. Exceptions to Lambertian reflection include water and ice surfaces.
However, the most frequently observed irradiance is the SW global irradiance (E G ) incident on a horizontal plane facing upward (pyranometer). E G is defined as E G = E I ·sin(γ s ) + E D , where γ s (degrees) is the solar elevation. Hence, measured E G and E D allow E I to be determined, the measurement of which is more difficult. There are algorithms available that break measured E G down into its fractions using a stochastic approach (see e.g., [31]).
E R can also be attributed to E G by E R = ρ s ·E G . ρ s is the surface albedo (0-1) that depends on ground surface properties. For a given location, ρ s can frequently be regarded as constant, but there may be seasonal variations [32], e.g., due to snow cover in winter.
Under overcast skies, E D is completely isotropic. However, under skies which are clear or only with low cloud cover, E D includes a significant anisotropic fraction that is concentrated around the solar disk with a view angle of ≤12 (degree). This circumsolar brightening originates from Mie scattering caused by aerosols and cloud particles that has a narrow peak in the forward direction. This anisotropic circumsolar fraction (f cs ) of E D is never available from operational observations. Hence, it must be derived using sky radiance models, e.g., that of Perez et al. [23], which accounts for γ s , sky clearness (overcast to clear) and sky brightness (cloud thickness and/or aerosol loading ranging from low to high). f cs is especially important if the irradiance on tilted surfaces is calculated (see e.g., [32]) because this depends on the azimuth of the radiation source, e.g., for an upright human.
The amount of radiation which each body emits depends on its (surface) temperature. Hence, the atmosphere and the ground are sources of electromagnetic radiation due to their temperature in the thermal-spectral range. The effective LW flux densities (W/m 2 ), measured by pyrgeometers, are

•
The isotropic diffuse downward thermal irradiance of the atmosphere (E A ) incident on a horizontal plane facing the upper half space; • The upward isotropic diffuse irradiance of the lower half space (E E ) incident on a horizontal plane facing the lower half space. E E includes the small fraction (≤5%) of ground-reflected E A .

Klima-Michel-Modell
The calculation of T mrt inside the Klima-Michel-Modell (KMM) [12] is based on Equation (1) for outdoor use. Breaking the unidirectional irradiances originating from the area surrounding the solar disk down into a vertical and a horizontal component the respective angle factor F p−N (Equation (1)) can be shown to equal f p (γ s ), because the weight in F p−N that accounts for the relative position of surface N to the person is in this case unity. For the standing posture of a person rotationally symmetric to its vertical axis a parameterization for f p based on diagrams in [6] can be applied ( Figure 1) [12]: Atmosphere 2020, 11, x FOR PEER REVIEW 5 of 22 symmetric to its vertical axis a parameterization for fp based on diagrams in [6] can be applied ( Figure  1) [12]: 0.308 cos 0.998 50000 (2) . Figure 1. Projected area factor fp,: both Klima-Michel-Modell (KMM) [12] and Park and Tuller [11] for the reference subject and additionally for a cylinder and a globe.
For a surrounding free of significant horizon limitations [33], Tmrt can, therefore, be derived directly from measured radiant flux densities, active radiometry, referred to below by "r" in subscripts. Because the view angle of the circumsolar fraction of ED is small, it can reasonably be treated like EI as originating from the solar disk [24]. Then, the equation of Jendritzky et al. [12] for the calculation of Tmrt, which assumes fcs = 0, can be extended for all fcs. It reads: σSB = 5.67•10 −8 (W•m −2 •K −4 ) is the Stefan-Boltzmann constant. Because isotropic diffuse radiation from one of two half spaces irradiates the body with the same intensity over all azimuth and elevation angles (Equation (1)), the integration over all angle factor Fp−N of this half space is fa = 0.5, where fa is independent of γs and the shape of the irradiated body [12,24,34]. Hereafter, the procedure based on Equation (3) is called Tmrt,r,KMM, relating it to the Klima-Michel-Modell (KMM) [12].

Six-Direction Instrument
Another method for determining Tmrt using observed irradiances is the six-directional instrument (Tmrt,r,6-Dir (K)) that posits the subject concerned as a rectangular solid [1,15]. This Tmrt,r,6-Dir (K) is also defined as the plane radiant temperature [8,9], however using pyranometers and pyrgeometers facing  [12] and Park and Tuller [11] for the reference subject and additionally for a cylinder and a globe.
For a surrounding free of significant horizon limitations [33], T mrt can, therefore, be derived directly from measured radiant flux densities, active radiometry, referred to below by "r" in subscripts. Because the view angle of the circumsolar fraction of E D is small, it can reasonably be treated like E I as originating from the solar disk [24]. Then, the equation of Jendritzky et al. [12] for the calculation of T mrt , which assumes f cs = 0, can be extended for all f cs . It reads: is the Stefan-Boltzmann constant. Because isotropic diffuse radiation from one of two half spaces irradiates the body with the same intensity over all azimuth and elevation angles (Equation (1)), the integration over all angle factor F p−N of this half space is f a = 0.5, where f a is independent of γ s and the shape of the irradiated body [12,24,34]. Hereafter, the procedure based on Equation (3) is called T mrt,r,KMM , relating it to the Klima-Michel-Modell (KMM) [12].

Six-Direction Instrument
Another method for determining T mrt using observed irradiances is the six-directional instrument (T mrt,r,6-Dir (K)) that posits the subject concerned as a rectangular solid [1,15]. This T mrt,r,6-Dir (K) is also defined as the plane radiant temperature [8,9], however using pyranometers and pyrgeometers facing up-down, left-right, forwards-backwards and belongs so to active radiometry. The six instruments of each type may be replaced by one that can be rotated in the six directions to deliver a temporal sequence of observations. The six weighting factors (W i ) for the observed SW (K i ) and LW (L i ) radiant flux densities (W/m 2 ) are derived using Fanger's formula [6] for f p of a rotationally symmetric person who is standing. W i equals 0.06 for up and down and 0.22 for the four cardinal points. The energy absorbed by the rectangular solid (S r )

Black Globe Thermometer
T mrt can be derived indirectly by solving the energy balance equation of a measuring system that is affected by convection, conduction and radiation (passive radiometer). T mrt is derivable when the system is in equilibrium, i.e., the radiant energy input is exactly balanced by convective heat dissipation [16]. The black globe thermometer is most frequently used for this [17]. This is essentially a hollow spherical globe containing an ordinary thermometer. The bulb of this thermometer is at the center of the sphere and theoretically represents the mean surface temperature of the sphere's inner surface. This integral value is composed of individual values of SW and LW radiant flux densities of the actual environment that vary with the solid angle. However, they may be smoothed, dependent on the heat conductivity of the globe's material.
The usual form of the black globe thermometer (BG) consists of a thin-walled (≈0.4 mm) copper sphere of diameter (d g ) of 0.15 (m) that is coated in matt black paint, emissivity ε g = 0.95 [8,22]. When exposed to radiation, such black globe thermometers usually register a higher temperature, the globe temperature (T g (K) or t g ( • C)), than an ordinary, radiation-protected thermometer that results in t a . Generally, the transient heat balance of a globe at time t is [35,36]: Q r (t) and Q c (t) are the radiant and the convective heat transfer rates, respectively, evaluated for the outer and for the inner surface of the globe. The sphere's surface (A g ) is 4·π·d g 2 (m 2 ), and δ g (m) is the shell's thickness. ρ (kg/m 3 ) is the mass density of the shell's material, and c (J K −1 kg −1 ) its specific heat capacity. dt is a time interval. This balance equation neglects heat conduction across the surface. This is only justified for metallic (high thermal diffusivity) shells that are thin relative to their overall dimensions [35]. Over the whole globe, the heat loss from the inner surface is zero because elements of the inner globe radiate out to each other and internal, free convection air currents redistribute heat from one point to another [35,37]. However, the radiant properties of the inner surface and the mass of the thermometer's bulb affect the time constant of the instrument. The energy balance on the globe's outer surface yields [8,22,35,38,39]: h c,g (W·m −2 ·K −1 ) is the convective heat transfer coefficient averaged over the surface of the sphere. In equilibrium, the right hand side of Equation (6) equals zero, lim t→∞ (Q r (t) − Q c (t)) outer = 0, and, then, Outdoor forced convection dominates for v a ≥ 0.03·(T g −T a ) 0.42 [38]. According to the similitude theory h c,g = κ·Nu/L . Nu (dimensionless) is the mean Nusselt number, i.e., the ratio of mean convective h c,g to conductive heat transfer κ (W·m −2 ·K −1 ) in the film layer, here at the boundary between the air and the globe's shell. L (m) is the characteristic length, for the sphere it is d g . Nu is additionally a function of v a on the sphere and of air properties, namely kinematic viscosity, specific heat capacity, thermal conductivity and density that are usually summarized in the value 6.3 [8,22] for standard atmospheric conditions. Thus, the black globe equation for forced convection (6) reads

Data
In order to realistically compare T mrt,r,6-Dir against T mrt,r,KMM , observations are required that cover in particular a sufficient number of solar altitudes and weather conditions. Such large amounts of observations are not available under all weather conditions for the 6-Dir instrument. Therefore, the fact is exploited that direct, diffuse and surface reflected SW and thermal LW irradiances onto tilted surfaces can readily be calculated based on (solar) geometry from standard meteorological irradiance measurements (see e.g., [32]). The sole exception in the transposition to tilted surfaces is the anisotropic fraction in E D . No such measurement is available and, thus, has to be modeled. To this end, a three component treatment of sky diffuse irradiance is applied based on known values of E D , E G and γ s . [23,25]. The circumsolar fraction f cs is then treated like E I , i.e., dependent on solar geometry. The small band of horizon brightness is added to the isotropic fraction in E D . E R is regarded as exclusively isotropic diffuse, like E A and E E . Hence, for an environment with no significant horizon limitations [33], the SW and LW flux densities on the vertical surfaces of the 6-Dir instrument can be derived from standard horizontal and beam irradiance measurements based on more or less exclusively geometric relations.
T mrt is applied as input for the calculation of thermal indices that represent steady state thermophysiological values of a reference person. Because steady state is usually achieved within 2 h [7,40], hourly sums of irradiation are the appropriate radiant input to characterize the radiant conditions in the adaptation phase leading to steady state.  [41]. All BSRN sites fulfill the conditions that they are free from immediate local effects such as pollution, heat island effects, local orography such as surface tilt and elevate horizon, and isolated atmospheric conditions such as local cloud developments [13].
The missing E R for LER and SBO is replaced by assuming ρ s = 0.2. The missing E E for both sites is modeled using a Bowen approach in solving the energy balance at the earth's surface [12]. The latter requires auxiliary quantities from synoptic meteorological observations (SYNOP) that are only available in very limited quantities for SBO. Because these are limited to 06 and 18 UTC, radiant components for high solar altitudes (noon) cannot be modeled.
For the years 2001 to 2003, hourly radiant components for Rostock-Warnemuende (RWA), Germany: 54.18 • N, 12.08 • E, 4 m MSL are also available ( Table 1). Their accuracy is in line with long-term measurements of national meteorological services [41]. The measuring field is located in the dunes bordering the Baltic Sea. Here, the SW E R is measured accounting for the higher albedo of the dune sand and the LW components E A and E E are measured based on Schulze pyrradiometers. Compared to pyrgeometers, the measurements should show approximately identical means, but the individual values are characterized by a slightly increased variance [42]. The SW components of the upper half space are measured at roof height of the nearby building of the meteorological site.
At RWA, T g from a black globe thermometer is also available, as are t a and v a taken at the height of the black globe. t g and t a are quantified with copper-constantan thermocouples according to the standard TGL 32603/2 [43], where t a is additionally controlled against the air temperature measured with a standard PT-100 resistance thermometer [44]. v a is taken with a Thies opto-electric anemometer, start-up-speed 0.3 (m/s). 10 min averages in UTC are accumulated to hourly means in true solar time.

Assessment Methodology and Reference
ISO 7726 [8] specifies the accuracy requirements for measuring methods based on the extent of the thermal annoyance being assessed. The more stringent requirements are for methods that relate to measurements carried out in moderate environments approaching comfort conditions, the less stringent are for environments subject to greater thermal stress. The latter may be considered for outdoor applications. Slightly simplified and thereby fractionally intensified in the extremes, the accuracy requirements of ISO 7726 [8] in T mrt are ±2 K for comfort and ±5 K for thermal stress. Subsequently, methods for the calculation of T mrt are tested against a reference method. It is assessed whether the differences in T mrt against the reference is within the thresholds ±2 K or ±5 K. In order to comply with the ISO standard, the maximum error probability (α 2K , α 5K ) has to be less than 5%. Further statistical values that are provided are the bias (K) as the mean of the differences, the root mean square error RMSE (K), and for linear regressions intercept (a) and slope (b), the coefficient of determination (R 2 ) and the standard error (SE). SE is a measure of scattering of the sample points related to the regression value.
The assessment methodology requires a reference method. All three methods can be traced to proposals of ISO 7726 [8] for the measurement/calculation of T mrt : • T mrt,r,KMM : Calculation from the temperature of the surrounding surfaces (Equation (1)) in the form for outdoor use [1,12]. f p (Equation (2)) parameterized from diagrams [6]. This parameterization is confirmed by more recent measurements (Figure 1) [11]. In the extended form T mrt,r,KMM accounts for f cs . T mrt,r,KMM can be calculated based on standard meteorological radiant measurements. • T mrt,r,6-Dir : Calculation from plane radiant temperatures [8] in the form for outdoor use [1,15]. The reference person is slightly simplified by a rectangular solid (Figure 1). The method implicitly accounts for f cs , because measured irradiances incident on vertical planes are used that depend on the azimuth of a plane relative to the azimuth of the solar disk. Variations in the accuracy of the instrument have been observed [45][46][47]. • T mrt,Tg,BG : Derived based on measured t g , t a and v a (passive radiometry). ISO 7726 qualifies T mrt,Tg,BG to be an approximation due to the difference in shape between a person and a globe. The vertical component in E I will be over-estimated in relation to that received by a standing person. The SW absorptivity significantly differs from that of the outer surface of a clothed person.
T mrt,r,KMM is adopted as the reference because it applies directly observable radiant components (subscript: r) that are used as standard in meteorology. In T mrt,r,6−Dir it is implicitly accounted for the circumsolar fraction of E D . However, this advantage cannot be completely used, because the data must be modeled (Section 2.5). Furthermore, f p for T mrt,r,KMM is derived from precise measurements of the real human bodies [6,10,11], while in T mrt,r,6-Dir the shape of the human body is slightly simplified to a rectangular solid.

Six-Direction Measurements
Variations in the accuracy of 6-Dir measurements have been observed [45][46][47]. Because the 6-Dir method assumes the subject to be a rectangular solid, its vertical cross-section for unidirectional SW irradiance is a function of the difference in azimuth between the solar azimuth and the azimuth angle of one of the vertical surfaces (∆A az (degrees)) that are irradiated by the unidirectional fraction (1 − Df ) in E G [30]. Df = (1 − f cs )·E D /E G is the isotropic diffuse fraction. The unidirectional cross-section is smallest for ∆A az = 0 • because there is only one sunlit vertical surface, and greatest for 45 • . In contrast, the vertical cross-section of a circular cylinder is azimuthally independent. Figure 1 presents the projected area factor of a cylinder that has the same relation of the cylindrical surface area and the surface area of the two parallel planes as the total vertical to the horizontal areas of the rectangular solid. Compared to that cylinder, the rectangular solid deviates in received irradiance due to the anisotropic fraction in SW irradiance (∆E an (W/m 2 )) indicated by: Because they are based on the same measured radiant fluxes, differences in T mrt,r,6-Dir and T mrt,r,KMM can only arise from the anisotropic radiant parameters and the slightly differing f p (see Figure 1). Therefore, the evaluation of differences is restricted to daylight hours. Figure 2a,b summarize the results for two azimuthal orientations of the 6-Dir instrument, where (a) assumes a measuring routine in which one of the vertical surfaces is set facing south, and (b) assumes that one of the vertical surfaces faces the solar azimuth. The reference accounts for variable f cs , despite the fact that it has to be modelled based on measurements. Figure 2c shows the effect of a reference that does not account for the circumsolar fraction in E D , i.e., in Equation (3) f cs = 0, as was originally suggested [12].  Figure 3 highlights the effect of circumsolar anisotropy in ED, fcs = variable, for determining Tmrt,r,KMM by comparing it with the case in which it is neglected, fcs = 0 [12]. In the extremes, Tmrt,r−fcs=var,KMM can deviate from Tmrt,r−fcs=0,KMM by approx. +5 (K) for solar altitudes γs close to the horizon and by approx. −7 (K) for the maximal γs of about 68 (degrees) within the data (due to SYNOPs only for 06 and 18 UTC at SBO (Table 1), higher solar elevations using modeled components are not attainable). Because the reference person is standing, the area projection factor fp for unidirectional SW irradiance increases with decreasing γs (Figure 1) and provides evidence for the positive deviations. For a given γs, the scattering in the differences arises from variable cloudiness due to the decreasing anisotropy effect with increasing cloud levels. At γs ≈ 32 (degrees), the optical air mass for the direct solar beam equals the mean value of the isotropic diffuse fraction and, thus, the anisotropic part in ED contributes to Tmrt,r,KMM by the same amount as if it were treated as isotropic. Because the optical air mass decreases in proportion to the increasing γs, there is a lower probability of the direct solar beam being scattered in its transfer to the earth's surface. Hence, for high γs the differences tend towards a higher absolute value. All orientations and assumptions remain within the ISO 7726 thresholds for thermal stress of ±5 (K) ( Table 2). However, the comfort threshold of ±2 (K) is met only by the cylinder and with an error probability ≤1.3% by the 6-Dir instrument facing south. Because of its azimuthal dependence, the scattering in the t mrt differences is stronger for the rectangular solid than for the cylinder (Figure 2a,d). It significantly increases if the reference does not account for variable f cs (Figure 2c versus Figure 2a). If a measuring routine is assumed in which one of the vertical surfaces faces the solar azimuth (Figure 2b), this results in a significant negative bias of approx. −1 (K), because only one of the vertical surfaces is sunlit with unidirectional irradiance (Table 2). Thus, the bias due to the azimuthal error approaches zero if the instrument has a fixed orientation. The statistics of Table 2 include the data of the sites LER and SBO where the upward SW and LW components are modeled. The statistical results do not differ if the data of the two sites are not accounted for. Table 2.
Six-directional (6-Dir) instrument and cylinder: Statistics for the difference T mrt,r,6-Dir -T mrt,r,KMM and error probability (α 2K , α 5K ) for compliance with ISO 7726; n = 29,833 daylight hours. The third order polynomials in Figure 2 prove the impact of the more greatly simplified assumptions concerning the human body compared to f p of Equation (2). Especially for increased T mrt,r,KMM , the differences tend toward values greater than zero. Here, the higher values are frequently combined with increased γ s and, thus, with a higher probability of anisotropic SW irradiances. The upper surface of the rectangular solid as well as the cover of the cylinder represents head and shoulder girdle of a human less precisely than Fanger's f p . As a consequence of the azimuthally dependent conditions of irradiance, the RMSE of the differences is almost double that in the rectangular solid compared to the cylinder ( Table 2). In contrast, standard radiation measurements apply horizontal receivers that are by definition azimuthally independent. This gives additional support to the selection of T mrt,r,KMM as the reference. Figure 3 highlights the effect of circumsolar anisotropy in E D , f cs = variable, for determining T mrt,r,KMM by comparing it with the case in which it is neglected, f cs = 0 [12]. In the extremes, T mrt,r−f cs=var,KMM can deviate from T mrt,r−f cs=0,KMM by approx. +5 (K) for solar altitudes γ s close to the horizon and by approx. −7 (K) for the maximal γ s of about 68 (degrees) within the data (due to SYNOPs only for 06 and 18 UTC at SBO (Table 1), higher solar elevations using modeled components are not attainable). Because the reference person is standing, the area projection factor f p for unidirectional SW irradiance increases with decreasing γ s ( Figure 1) and provides evidence for the positive deviations. For a given γ s , the scattering in the differences arises from variable cloudiness due to the decreasing anisotropy effect with increasing cloud levels. At γ s ≈ 32 (degrees), the optical air mass for the direct solar beam equals the mean value of the isotropic diffuse fraction and, thus, the anisotropic part in E D contributes to T mrt,r,KMM by the same amount as if it were treated as isotropic. Because the optical air mass decreases in proportion to the increasing γ s , there is a lower probability of the direct solar beam being scattered in its transfer to the earth's surface. Hence, for high γ s the differences tend towards a higher absolute value.

Black Globe Thermometer
A total of 25311 hourly mean values of measured Tg of a black globe is available for Warnemuende (RWA). They also include hours with precipitation that may impact Tg due to evaporative effects etc. Based on hourly synoptic observations, all Tg means are excluded that, in the present weather or within the past few hours, showed WMO-FM12 code numbers [48] greater than 10, i.e., cases with drizzle, rain, snow, thunderstorm etc. Furthermore, hours are excluded with an air speed at the height of the globe thermometer which is less than that of forced convection and/or with

Black Globe Thermometer
A total of 25311 hourly mean values of measured T g of a black globe is available for Warnemuende (RWA). They also include hours with precipitation that may impact T g due to evaporative effects etc. Based on hourly synoptic observations, all T g means are excluded that, in the present weather or within the past few hours, showed WMO-FM12 code numbers [48] greater than 10, i.e., cases with drizzle, rain, snow, thunderstorm etc. Furthermore, hours are excluded with an air speed at the height of the globe thermometer which is less than that of forced convection and/or with a difference in the air temperature registered by the thermocouples and the PT100 thermometer that exceeds 2 (K). Thus, 9191 values remain that are all covered by measured radiant components required to calculate T mrt,r,KMM .
t g is not a thermodynamic property [49] but a measure of the complex radiant and convective effects of the environment on the black globe. Figure 4 examines some of them and their implications for the derived T mrt,Tg,BG . It is also closely related to frequent attempts to predict t g based on standard meteorological measurements [34,[49][50][51][52][53][54]. Figure 4a relates t g ( • C) to t a ( • C), where t g is split into three bins of radiant conditions: (i) at nighttime, i.e., sun is below the horizon, at daytime dependent on the value of E I : (ii) cloudy, i.e., E I < 120 (W/m 2 ), no sunshine according to WMO [14], and (iii) sunny, i.e., E I ≥ 120 (W/m 2 ).   During the daytime, SW irradiation transfers heat onto the black globe, and the absorbed SW energy very often results in an increased tg, the amount of which depends on the SW intensity. At daytime in cloudy hours the sky is most frequently overcast, i.e., ED accounts for the largest share in EG. Compared to nighttime, the absorbed SW irradiance gives reason that the bias turns to positive and that the slope of the regression increases to 1.13. Compared to nighttime, the standard error is increased by a factor three and amounts to 1.9 (K). In the sunny hours, there is stronger SW heat transfer onto the black globe, the slope increases to 1.18, at +7.1 (K) the regression has a markedly positive bias and the standard error increases to +2.9 (K) ( Table 3). The scattering in the data is significantly increased compared to the nighttime because va impacts on the convection, but tmrt also At nighttime the radiant heat transfer is limited to E A and E E , the thermal spectral band. Then, t a and t g are closely and linearly regressed with the slope of the regression line close to unit ( Table 3). The difference t g -t a is negatively biased by −0.9 (K), which provides evidence that, in most cases, it is the lower half space that in balance transfers radiant heat to the atmospheric environment. The scattering of the data can be attributed predominately to the varying v a that is responsible for the convectional heat loss. During the daytime, SW irradiation transfers heat onto the black globe, and the absorbed SW energy very often results in an increased t g , the amount of which depends on the SW intensity. At daytime in cloudy hours the sky is most frequently overcast, i.e., E D accounts for the largest share in E G . Compared to nighttime, the absorbed SW irradiance gives reason that the bias turns to positive and that the slope of the regression increases to 1.13. Compared to nighttime, the standard error is increased by a factor three and amounts to 1.9 (K). In the sunny hours, there is stronger SW heat transfer onto the black globe, the slope increases to 1.18, at +7.1 (K) the regression has a markedly positive bias and the standard error increases to +2.9 (K) ( Table 3). The scattering in the data is significantly increased compared to the nighttime because v a impacts on the convection, but t mrt also varies.
In the next step, a T mrt,r,BG is simulated for a black globe based on measured radiant components. Because of the sphere's simple geometry, its area projection factor is f p = 0.25 and independent of γ s (Figure 1). Then, T mrt,r,BG can be derived by applying Equation (3), assuming α sw,Sk = 0.95 and ε sw,Sk = ε g =0.95 [8]. Because it is based in this case on measured radiant quantities, T mrt,r,BG is assumed as the reference. By highlighting the difference to T mrt,r,BG, Figure 4b shows that T mrt,Tg,BG significantly underestimates T mrt,rKMM , bias −1.9 (K), which in this study is defined as reference ( Table 4). The ISO 7726 [8] requirements for neither the comfort class nor thermal stress are met. The differences scatter significantly, especially for values of T mrt,r,BG greater than approx. +20 ( • C).  Figure 5a relates the difference of simulated T mrt,r,BG to the reference T mrt,r,KMM , where both T mrt,r,BG and T mrt,r,KMM use the same measured radiant components. The difference increases sharply with rising T mrt,r,KMM because the SW absorption of the black globe is α sw,Sk = 0.95 compared to KMM of α sw,Sk = 0.70. There is noticeable data scatter around their 4th order polynomial, SE = 1.68 (K), due to the fact that f p of KMM is a function of γ s while this is not the case for a sphere. Because of the enhanced SW absorption, the simulated T mrt,r,BG does not meet the ISO requirements, either for the comfort or for the stress class (Table 4). SW absorption, the simulated Tmrt,r,BG does not meet the ISO requirements, either for the comfort or for the stress class (Table 4). This alters when simulating a Tmrt,r,GG for a gray globe (GG), assuming αsw,Sk = 0.70 as for KMM (Figure 5b, Table 4). The fp-related scattering around the second-order polynomial remains unchanged; it is, however, somewhat reduced, SE = 1.25 (K). Because the simulated Tmrt,r,GG complies with the ISO requirements both for the comfort and for the stress class (Table 4), it becomes evident that the fp dependence is only of minor importance compared to the SW absorption. Time periods with summery more or less cloudless skies around noon may be characterized by EI > 600 (W/m 2 ).
Then, α5K is just beyond compliance with the ISO 7726 stress class due to the differences in shape between a globe and a standing person (Table 4).  This alters when simulating a T mrt,r,GG for a gray globe (GG), assuming α sw,Sk = 0.70 as for KMM (Figure 5b, Table 4). The f p -related scattering around the second-order polynomial remains unchanged; it is, however, somewhat reduced, SE = 1.25 (K). Because the simulated T mrt,r,GG complies with the ISO requirements both for the comfort and for the stress class (Table 4), it becomes evident that the f p dependence is only of minor importance compared to the SW absorption. Time periods with summery more or less cloudless skies around noon may be characterized by E I > 600 (W/m 2 ). Then, α 5K is just beyond compliance with the ISO 7726 stress class due to the differences in shape between a globe and a standing person (Table 4). Figure 5c compares T mrt,Tg,BG to T mrt,r,KMM : The scattering of the data values around their polynomial fits, SE = 4.66 (K), is considerably greater compared to T mrt,r,BG (Figure 5a), SE = 1.68 K). In total, however, the values of the T g -derived T mrt,Tg,BG are lower than those of T mrt,r,BG . This fact is clearly demonstrated by the two polynomial fits, where the dashed line is the fit of Figure 5a. Overall, T mrt,Tg,BG does not comply with the ISO 7726 requirements, either for the comfort or for the thermal stress class (Table 4). Splitting the data into nighttime values and the daytime data into cloudy and sunny components reveals that the ISO 7726 class for thermal stress is met at nighttime (Table 4). There is a slight bias in the data, −0.27 (K), RMSE = 1.89 (K). During the daytime, the data do not comply with the ISO 7726 requirements, there is a positive bias for cloudy conditions +0.83 (K), increasing to +3.79 (K) during sunny periods (Table 4). Thus, the daytime results of the globe instrument show significant shortcomings compared to those derived from active radiometry. These shortcomings are attributable to the SW irradiation E G that covers the isotropic diffuse fraction of E D and the unidirectional fraction E I as well as the circumsolar part of E D . Especially under sunny conditions and high radiant intensity the unidirectional fraction accounts for the largest share in E G with~75-85%. Thus, E I is suitable to characterize conditions where the black globe thermometer is subject to strong asymmetric SW irradiation. Plotted over E I , the daytime data of the difference t mrt,Tg,BG − t mrt,r,KMM show a bias of +2.2 (K) for 120 < E I ≤ 600 (W/m 2 ) due to the enhanced SW absorption of the black globe compared to KMM (Figure 5d, Table 4). Stronger asymmetric radiant conditions can be attributed to values of E I > 600 (W/m 2 ). Here the bias rises sharply with increasing E I (Table 4). For a gray globe that has the same radiant cross-section as the black globe but a reduced SW absorptance (α sw,Sk~0 .70), the maximally absorbed energy from unidirectional irradiances may be reduced to approx. 600 (W/m 2 ) and the bias will be closer to zero. Because of the comparable amount of absorbed unidirectional radiant heat, it can be inferred from Figure 5d that the scattering in the data of a gray globe will be approximately the same as for the black globe in the region E I < 600 (W/m 2 ), and, thus, significantly enhanced compared to T mrt,r,GG ( Figure 5b) and markedly too large to comply with the ISO 7726 stress class.

Discussion
At nighttime, T mrt,Tg,BG remains within the ISO 7726 [8] thresholds for the stress class when applied as a measure for T mrt,r,KMM . During daytime, and especially in summery sunny conditions, there is a significant positive bias and, in particular, markedly greater scattering in the values so that the stress class thresholds are far exceeded. This provides additional evidence that the increased scattering is a typical outdoor problem of a black globe thermometer, and coincides with other findings (e.g., [55]). Under direct sun, the SW irradiance is strongly asymmetric; a temperature difference of up to 7 (K) in opposing areas of the inner surface has been measured [56]. Additionally, the wind field on the outer border of the sphere's film layer is not entirely uniform [57]. Therefore, the wind direction relative to the hot spot of direct irradiation can influence the overall convectional cooling of the globe. This spatial relationship between the fields of irradiance and of wind is important for the internal convection. Both theoretically and in experiments, the internal flow and temperature fields become asymmetric [37,55,58]. Thus, under strong asymmetric irradiation the temperature at the sphere's center is unable to represent the average surface temperature of the globe thermometer system.
The black globe was originally introduced in order to measure radiant heat in relation to human comfort [17]. By chance, Nu of the applied sphere with d g = 0.15 (m) is identical to that of a naked recumbent man [57]. Based on the similitude theory, this diameter is fundamental when T g is related to human comfort, i.e., T g is closely correlated with the operative temperature. Nevertheless, in the case of the black globe, it has the disadvantage that its SW absorption is significantly higher than that of human skin/clothing and thus tends to overestimate T mrt when applied to human thermophysiology [22]. In contrast to comfort, the diameter is not predetermined using the black globe as a passive radiometer in order to determine T mrt [16]. According to Equation (8), smaller diameters will improve the convective heat loss but will demand more accurate measurements of air velocity and ambient temperature [8,22,59,60].
Rewriting the transient form of the globe's heat balance, Equation (6), in a linearized form allows, through temporal integration, the time constant of the instrument (τ (s)) to be determined for a stepwise change [38]: (Q Ir − Q c ) outer = ε g ·h r,g ·(T mrt − T g ) − h c,g ·(T g − T a ) = −ρ·c·δ·dT/dt, where h r,g ≈ 4·σ SB ·((T mrt + T g )/2) 3 (W·m −2 ·K −1 ) is the mean radiant heat transfer coefficient [61]. τ represents the time in which T g reduces by 63.2% to the value of equilibrium. Within the response time, i.e., six to seven times τ, T g has balanced out. Restricted to the heat exchange between the globe's shell and the environment, τ is a function of h c,g . This depends, among other things, on the globe's geometry and its material characteristics, as well as on h r,g . τ strongly decreases as the result of increasing h c,g that is a function of v a . Outdoors, v a is greater than in typical indoor conditions at v a ≤ 0.2 (m/s). At RWA, where the measuring field is located in the dunes bordering the Baltic Sea, the mean of all records is v a = 2.34 (m/s), the median 1.8 (m/s). In typical summery conditions, t a = 18.4 ( • C), v a = 2.6 (m/s), t mrt,Tg,BG = 25.8 ( • C), the calculated response time is 5.25 (min) and is lower than for the same conditions by a factor of~3, yet v a = 0.20 (m/s). However, this does not take into account the heat conductivity of the shell's material and especially not the radiant properties of the inner surface and the mass of the thermometer's bulb which also affect the time constant of the instrument. The mass of the applied thermocouple is small, the emissivity of the oxidized copper inside the sphere is, at~0.4, relatively low and there is only free convection. Hence, overall the instrument's response time will be increased, but should not exceed observed values of about 10 min for outdoor air velocities of 2 (m/s) [62]. Instruments have been constructed with a fan inside the sphere for forced convection in order to improve their response time [36,63]. However, they have not achieved greater acceptance.
Outdoors the environment is not in a steady state, i.e., the globe is impacted by short-term fluctuations, e.g., by wind gusts and brief cloud shadows in the case of broken cloud coverage, but usually this is only temporary. Hence, the hourly averages of measured T g will rarely represent a steady state in the globe thermometer. Over the course of the day, they may include a time shift compared to the radiances, however these can be regarded as more or less insignificant.
Indoor black globe systems have been introduced, the shell of which consists of a table-tennis ball (acrylic), diameter~0.04 (m), that is painted matt black. Their purpose is to achieve a quicker response time and lower instrument costs [64,65]. As a result of these reported advantages, such black globe instruments are now also being used outdoors [20,21]. However, under the strong asymmetric SW irradiation found outdoors, the heat conductivity of acrylic can no longer be ignored in terms of the heat balance of the system because it is lower than that of copper by a factor of~1000. This may outweigh the reduced response time due to the lower mass density of acrylic compared to copper, and the reduced diameter. It is not recommended to use table-tennis balls outdoors as black globe thermometers in order to determine T mrt because they yield unpredictable errors in the measurements [37,55].
The wet-bulb-globe-temperature (WBGT) index [18] is an index of thermal stress and has been adopted as standard ISO 7243 [19]. WBGT has the advantage that it can also be evaluated directly outdoors by taking readings of its three input components: the natural wet bulb temperature (t nw ( • C)) as well as t a ( • C) and t g ( • C). They are summarized in the weighted average WBGT = 0.7·t nw + 0.1·t a + 0.2·t g . The weights are empirically determined by linking them with heat casualties and work-rest cycles needed to prevent thermal strain on humans in the workplace [3,18,66,67]. T mrt is not explicitly required as WBGT input. Hence, uncertainties in t g as found in this study are irrelevant for the accuracy of the WBGT index because they are accounted for in the weights. However, when the diameter of the globe used differs from the standard 0.15 (m), the measured t g value has to be adapted to the standard configuration [35,50,68]. Here the algorithm is based on v a and the magnitude of the difference t g −t a [62]. It is difficult to measure t nw and t g , which are not thermodynamic properties [49]. Thus, measures have been undertaken in order to derive them from environmental data. In the case of t g , this can be achieved by solving Equation (7) for T g . The required input T mrt may be derived, assuming that it is a linear function of t a [51,52], i.e., radiant flux densities are only estimated. In more recent approaches, T mrt is explicitly derived from readily available standard meteorological data that include the above-mentioned data from active radiometry (e.g., [24,34,53,54,67]). Here, T mrt,r,BG can be calculated for a black globe by means of Equation (3) using f p = 0.25 and α sw,Sk = 0.95. Then, Equation (7) is solved for T g by iteration or by using empirical/polynomial methods. As this study found, it is advantageous for such empirical approaches to differentiate during the daytime between conditions in which the black globe is sunlit or is shaded [50] and to address separately nighttime data that are more closely linked to t a (Figure 4a) [54,69]. From the correlation of modeled against observed t g [49,50,54] and from the difference T mrt,Tg,BG − T mrt,r,KMM as a function of t g [70], it is apparent that the observed t g exhibits the same problem as seen here, especially under sunny conditions (see Figure 4a,b, Table 4).
With the exception of E I, observed standard radiant components refer to horizontally mounted receivers. These measurements have the advantage that they do not depend on the solar azimuth. In contrast, the irradiance on the vertically mounted receivers of the 6−Dir instrument depends on the solar azimuth and the measured values consist of radiant components incident from the lower and the upper half space. If the irradiance incident from the ground can be regarded as Lambertian equivalent, which is generally the case, then the sum of E I and the circumsolar fraction of E D can be derived from the irradiance difference of the sunlit and the shaded receivers accounting for γ s [47]. This implies, however, an environment free of significant horizon limitations [33]. The 6−Dir instrument is mainly intended for measurements in micro-structured urban environments. In, e.g., a narrow street canyon with the measuring site positioned closer to a building, the albedo of the building surfaces will significantly affect the measured diffuse SW irradiation, and the precondition for the method is violated.
In an urban environment, standard active radiometry combined with Equation (3) can also be used in order to derive T mrt for a human bioclimatological assessment. Deploying a sun tracker to measure E I or E D may be costly. An alternative could be to deploy a rotating shadow band irradiometer. After correction for its deviation from the Lambertian response and for its nonuniform spectral response, the instrument delivers both E G and E D in moderate to good quality, which allows E I to be inferred [71,72]. Furthermore, the input values for Equation (3) may be measured at roof level, i.e., without horizon limitations [33], or taken from a nearby meteorological site. Then, the derived T mrt,r,KMM is valid for the sky fraction seen from the point of assessment, the sky view factor, and the observed radiation flux densities can serve as a force in modeling the radiation transfer within the complex urban environment [21,[73][74][75]. The observations of irradiation at meteorological sites frequently do not include all the components required to derive T mrt,r,KMM . It has been shown that it is possible to calculate it with the required accuracy for the ISO 7726 thermal stress class if E G and a synoptic observation are available to serve as the base in order to model the missing radiant components [30,76].
As part of the model output, Numerical Weather Prediction (NWP) usually delivers in hourly resolution not only t a , e a (2 m above ground) and v a (10 m above ground) but also gridded data of irradiation sums starting with the initialized analysis. Based on the example of the Deutscher Wetterdienst (DWD), E G , E D , and the LW irradiation balance E A −E E may be available. The latter can be broken down using the forecast temperature of the ground surface or of the snow/ice surface, and converted using the Stefan-Boltzmann law to E E . The SW ground-reflected irradiation E R is derivable using the predicted ground albedo that varies over time, in particular due to snow cover or sea ice. Then, in NWP postprocessing, all input parameters are available to derive not only T mrt,r,KMM via Equation (3) but also thermal indices as a basis for forecasting the spatial and temporal development of thermal health hazards (see e.g., [77]). Furthermore, test reference years derived from gridded high resolution datasets can also be helpful in the context of planning and health issues [78,79].

Conclusions
Among the required input parameters for rational thermal indices, T mrt is the most complex. The quality of these indices and of the derived thermal assessment of the environment depends on the quality of the input. The requirements for T mrt are specified by ISO 7726.
Based on active radiometry T mrt accounts for the absorbed radiation from both the SW and LW spectral bands. Most of the radiant components are isotropic. Restricted to the SW band there are, however, also unidirectional irradiances: E I and under clear or partly obstructed skies a significant circumsolar fraction f cs in E D . f cs significantly impacts T mrt with noticeably reduced values for high