3.1.3. Emissivity Determination Using the Emissivity Box

We constructed a two-lid emissivity box at King's College London (Figure 3), based on the design of Rubio et al. [33,34] with (i) an improved outer thermal insulation layer surrounded by a robust outer case, (ii) a 3D-printed angled port to hold the radiometer at a constant view zenith angle of 5◦ to reduce the "Narcissus" effect, as in Göttsche and Olesen [37], (iii) continuous 1 Hz sampling of the radiometer measurements as in Göttsche and Olesen [37] to enable identification and rejection of erroneous readings (e.g., when conditions were not stable) during post-processing readings, and (iv) the addition of a "heating tray" to help the hot lid more quickly achieve its optimal temperature, while also reducing heat loss in cooler conditions. While our design could also be used for the one-lid approach, the two-lid method is considered more robust in windy or otherwise variable field conditions [34], some of which we encountered during our study.

**Figure 3.** (**a**) Insulated emissivity box with the cold base and battery-powered hot lid. Details include **Figure 3.** (**a**) Insulated emissivity box with the cold base and battery-powered hot lid. Details include (**b**) internal walls of highly polished aluminium, (**c**) a 3D printed 5◦ radiometer port for consistent off-centre angled sampling to avoid the "Narcissus" effect, and (**d**) evenly distributed electronic heating pads on the hot lid enabling heating up to at least 60 ◦C when combined with the heating tray.

μ ܶ ܮ Wmିଶsrିଵμmିଵ 10.55 μ Broadband surface emissivity was determined using our emissivity box via a sequence of BT measurements made with a Heitronics KT15.85 IIP radiometer fitted in the angled radiometer port to sequentially view the target surface and the base, as described in [31]. This radiometer is the same model as that used at the four permanent LST validation stations described in Göttsche et al. [50] and operates over the spectral range 9.6–11.5 µm, which is located well within the LWIR atmospheric window (Figure 4). BT measurements from the radiometer (*T*, kelvin) were converted into spectral radiances (*L*, Wm−<sup>2</sup> sr−1µm−<sup>1</sup> ) using Planck's radiation law (Equation (4)) evaluated at the effective radiometer central wavelength (approximately 10.55 µm; the exact value depending on the target temperature). Laboratory calibration tests confirmed the radiometer to have an absolute accuracy of

170 cm<sup>ଶ</sup>

±0.5 K plus 0.7% of the difference between the BT of the target and the radiometer body temperature (taken as the ambient temperature). For example, if the ambient temperature was 300 K and the target BT was recorded as 295 K, the absolute accuracy was determined to be ±(0.5 + [0.7/100 × 5]) = ±0.535 K. When fitted into the angled port, the observed surface area was 170 cm<sup>2</sup> .

**Figure 4.** Spectral response function of the Heitronics KT15.85 radiometer (blue, left axis) overplotted on the atmospheric transmission of a standard mid-latitude summer atmosphere (red, right axis) calculated using MODTRAN 5.0 [61].

ߝ To make the measurements necessary to estimate the targets surface emissivity, the box was first placed on the target surface sample and left for two minutes to ensure stabilised temperatures. Measurements then proceed as in Figure 5, with the rationale for this sequence explained in Rubio et al. [33]. Using the same nomenclature as in Rubio et al. [33] and Figure 5 the broadband emissivity (ε0) of the target surface if the box were ideal was calculated as:

$$
\varepsilon\_0 = \frac{L^3 - L^1}{L^3 - L^2} \tag{6}
$$

ܮ ଶ ܮ ଵ ܮ ଷ where (in order of measurement), *L* 2 is the sample radiance measured when the box is over the ground sample with the cold lid in use, *L* 1 the sample radiance made with the hot lid in use instead of the cold lid and *L* 3 the radiance obtained when putting the box with the hot lid on over a cold base with the same emissivity as the cold lid.

ߝߜ However, the box departs from non-ideal behaviour (because the emissivity of the hot lid cannot be 1 and the emissivity of the cold lid cannot be 0) as detailed by Rubio et al. [34], who developed a correction factor (δε) for these effects equal to:

$$\delta\varepsilon = (1 - \varepsilon\_0) \left( 1 - \frac{\left(L^3 - L^2\right)(1 - \varepsilon\_c)}{(L^3 - L^2) - (L^3 - L^1)P + (L^2 - B\_c)Q} \right) \tag{7}$$

ܲ ܳ <sup>ୡ</sup>ߝ)݂ = ܲ (ୡߝ)݃ = ܳ (୦ߝ , where ε<sup>c</sup> is the emissivity of the polished aluminium, the term *B*<sup>c</sup> the radiance measured through the box with the cold lid on top and cold base below (effectively equal to the blackbody spectral radiance at the temperature of the aluminium), and *P* and *Q* are box-specific parameters between 0 and 1 defined

ܲ ܳ

by the box geometry and material properties such that *P* = *f*(εc, εh) and *Q* = *g*(εc) (see [34] for exact expressions of *P* and *Q*).

ܮ <sup>ଶ</sup> ܤ<sup>ୡ</sup> **Figure 5.** Radiance measurement sequence for the two-lid variant of the emissivity box method, using nomenclature from Rubio et al. [33]. Measurements proceed from left (*L* 2 ) to right (*B*c).

.(0.98 = <sup>୦</sup>ߝ The final broadband emissivity estimate of the target sample surface is then given by the sum of the outputs from Equations (6) and (7):

$$
\varepsilon = \varepsilon\_0 + \delta \varepsilon \tag{8}
$$

 = ߝ (ୢT୰ୟ(ܤ <sup>୩୧୬</sup>ܶߪ ସ The emissivity of the hot lid is that of the high emissivity paint it was covered in (provided by the manufacturer as ε<sup>h</sup> = 0.98). Prior to field deployment, measurements were conducted to determine the emissivity of the cold lid as derived in Appendix 1 of Rubio et al. [34]:

$$\varepsilon = \frac{B(\mathcal{T}\_{\rm rad})}{\sigma T\_{\rm kin}^4} \tag{9}$$

where *B*(*T*) is the spectral radiance derived from Planck's radiation law at temperature *T* (kelvin) as in Equation (4) and <sup>σ</sup> is the Stefan–Boltzmann constant 5.67 × 10−<sup>8</sup> Js−1m−2K −4 . Through this method, the emissivity of the polished aluminium (εc) of the cold lid was determined as 0.05, resulting in P and Q as 0.0123 and 0.4223 respectively for the box deployed herein. These values were derived again based on measurements made at the end of each field campaign to identify any changes, associated for example with oxidisation of the box interior aluminium surface or damage to the paint of the hot lid, but no such changes were found.

Wmିଶsrିଵμmିଵ A minimum of five repeated measurements were collected per sample, with averages and standard deviations of the multiple measurements calculated. Uncertainties for each sample were taken as the standard deviation of the multiple measurements. Example values for the two-lid box method developed at King's College London are shown in Table 3.

 ࢿ ࢿࢾ ࢿ ܋ **Table 3.** Example values for the two-lid box measurements, with gravel, grass, and sand shown. Measured temperatures are expressed in kelvin and equivalent radiances in brackets in Wm−<sup>2</sup> sr−1µm−<sup>1</sup> .

