ABSTRACT
Temperature Measurement ................................................................................. 185 6.1.2 Relations and Databases for Spectral Emissivity ............................................. 190 6.1.3 Needs for Emissivity-Temperature Separation Methods ............................... 190
6.2 Single-Color and Two-Color Pyrometry ....................................................................... 192 6.2.1 Single-Color Pyrometry........................................................................................ 192 6.2.2 Two-Color Pyrometry .......................................................................................... 193
6.3 Multiwavelength Pyrometry ........................................................................................... 194 6.3.1 Interpolation-Based Methods .............................................................................. 195 6.3.2 Regularization by Using a Low-Order Emissivity Model .............................. 196
6.3.2.1 Description of Emissivity Models........................................................ 196 6.3.2.2 Least-Squares Solution of the Linearized ETS Problem ................... 197 6.3.2.3 A Look at the Solutions of the ETS Problem ..................................... 205 6.3.2.4 Least-Squares Solution of the Nonlinearized ETS Problem............. 207
6.4 Emissivity-Temperature Separation Methods in the Field of Remote Sensing ...... 213 6.4.1 Combined Atmospheric Compensation and Emissivity-Temperature
Separation............................................................................................................... 215 6.4.2 Separate Evaluation of the Atmospheric Parameters ...................................... 217 6.4.3 Emissivity-Temperature Separation Methods.................................................. 218
6.4.3.1 Normalized Emissivity Method........................................................... 218 6.4.3.2 Temperature-Emissivity Separation (TES) Method.......................... 219 6.4.3.3 Spectral Smoothness Method (SpSm).................................................. 220 6.4.3.4 Multi-Temperature Method.................................................................. 222
6.5 Conclusion.......................................................................................................................... 226 Nomenclature ............................................................................................................................. 227 References.................................................................................................................................... 228
6.1.1 Basic Relations for Sensed Radiance in Radiative Temperature Measurement
Matter spontaneously emits electromagnetic radiation in a very broad spectrum enclosing ultraviolet (UV), visible light, infrared (IR), and microwaves. The emitted radiance from a surface in a given direction depends on wavelength, temperature, and the considered
matter and direction. For a solid material, it also depends on the surface state: presence of corrosion and roughness. The maximum emitted radiance is given by Planck’s law. It only depends on wavelength and temperature (Siegel and Howell 1972):
B(l,T) ¼ C1 l5
1 exp (C2=lT) 1 (6:1)
B(l,T) is expressed in W=m3=sr, wavelength l in m, and temperature T in K, with C1¼ 1.191 1016 W m2 and C2¼ 1.439 102 m K (see Figure 6.1). B(l,T) is also called the blackbody radiance. A blackbody surface absorbs all incoming
radiation, and no other surface, at the same temperature, emits more thermal radiation than it does. The blackbody is essentially a thermodynamic concept and it is difficult to find a material presenting such properties over the entire electromagnetic spectrum. The blackbody radiance is described in Figure 6.1 for different temperature levels. The
maximum emission is observed at a wavelength lmax such that lmax T¼ 2898 mm K, which is Wien’s displacement law. The peak emissive intensity shifts to a shorter wavelength at a higher temperature in inverse proportion to T. A common approximation to Plank’s law is Wien’s law, which is also plotted in
Figure 6.1:
BW(l,T) ¼ C1 l5
exp C2 lT
(6:2)
The approximation error increases with the wavelength. One can, however, consider that Wien’s approximation is valid in the rising part of the radiance curve. As a matter of fact, the error is less than 1% provided lT< 3124 mm K. It is obvious that by measuring the thermal radiation emitted by the blackbody surface at
a given wavelength and with reference to Planck’s law, one can infer its temperature. This idea is at the origin of pyrometry, thermography, microwave radiometry, and more
generally all electromagnetic-based approaches that rely on the thermal radiation intensity measurement for temperature characterization. The sensitivity of blackbody radiance to temperature, according to Planck’s law, is
plotted in Figures 6.2 and 6.3. Figure 6.2 refers to absolute sensitivity qB=qT whereas Figure 6.3 refers to relative sensitivity B1qB=qT. The absolute sensitivity presents a maximum at a wavelength such that lT¼ 2410 mm K. For a blackbody at 300 K, maximum radiance is observed at l¼ 9.65 mm; however, the maximum sensitivity to temperature variations is observed at a shorter wavelength, namely, l¼ 8.03 mm. On the other hand, the relative sensitivity is continuously decreasing (see Figure 6.3). The trend is like 1=l at short wavelengths. The decreasing nature of relative sensitivity would favor short wavelengths for temperature measurement. Actually, one should consider all three aspects: radiance level, absolute sensitivity, and relative sensitivity, together with the spectral detectivity
spectral band for temperature measurement. The ratio between L(l,T, u,w), the radiance effectively emitted by a surface in the
direction (u,w), and the blackbody radiance at same wavelength and same temperature is called the emissivity:
e(l,T, u,w) ¼ L(l,T, u,w) B(l,T)
1 (6:3)
The emissivity generally depends on the surface temperature but, just for convenience, we will drop the T dependency. Second Kirchhoff’s law states that the emissivity in a given direction is equal to the
absorptance in the same direction:
e(l, u,w) ¼ a(l, u,w) (6:4)
The energy conservation law for an opaque material (i.e., the energy that is not absorbed by the surface is reflected in all directions) leads to the following relation between absorptance and directional hemispherical reflectance:
a(l, u,w)þ r0\(l, u,w) ¼ 1 (6:5)
The radiation that leaves the surface L(l,T, u,w) is the sum of the radiation emitted by the surface and the reflection by the surface of the radiation coming from the environment in all directions (ui,wi) of the upper hemisphere:
L(l,T, u,w) ¼ e(l, u,w)B(l,T)þ ð 2p
r00(l, u,w, ui,wi)L #(l, ui,wi) cos uidVi (6:6)
where r00(l, u,w, ui,wi) is the bidirectional reflectance. Let us now consider temperature measurement with an optical sensor. Depending on the
application, the sensor is at a distance ranging from a fraction of a meter, in common industrial processes, to several kilometers in the case of airborne remote sensing and up to hundreds or even thousands of kilometers in the case of satellite remote sensing. Apart from the cases based on vacuum operation, the sensed thermal radiation is thus transmitted through an air layer ranging from a few centimeters to the whole atmosphere thickness (air layer thickness can be higher in the case of near-horizontal line of sight). Along this optical path, only a fraction of the radiation is transmitted (the corresponding fraction is defined by the transmission coefficient t(l, u,w)). The self-emitted radiation of the air layer between the surface and the sensor, L"(l, u,w), finally adds to the transmitted fraction to give the at-sensor radiance Ls(l,T, u,w):
Ls(l,T, u,w) ¼ t(l, u,w)L(l,T, u,w)þ L"(l, u,w) (6:7)
A common approximation is to consider that the surface is Lambertian, i.e., its optical properties are direction independent. Equation 6.6 is then simplified as follows:
L(l,T) ¼ e(l)B(l,T)þ (1 e(l))E #(l,T) p
(6:8)
E#(l,T) ¼ ð 2p
Lenv(l, ui,wi) cos uidVi (6:9)
By introducing L#(l,T)¼E#(l,T)=p, the equivalent isotropic environment radiance, one gets
L(l,T) ¼ e(l)B(l,T)þ (1 e(l))L#(l) (6:10)
The influence of the air layer between the surface and the sensor was expressed through its transmission and its self-emission. The same approach can be applied to model the influence of the collecting optics of the sensor. Combining all together, a global transmission and a global self-emission can be defined therefrom. This development has shown that, even for Lambertian surfaces, the sensed radiation
depends on a series of additional variables: the surface emissivity, the irradiance from the environment, the path transmission, and the path self-emission. Therefore, in order to get the target temperature from the measured radiance, one also has to estimate these variables. Depending on the application, the difficulties they introduce are very different:
1. Pyrometry of High-Temperature Surfaces Generally the sensor is at a close range (the air path is on the order of 0.1-10 m). Therefore, by carefully selecting the wavelength(s), the air transmission can be very high. At the same time, the air self-emission can be negligible. In any case, a calibration can be performed for correcting the optical path transmission and its self-emission by aiming a blackbody which is put at the same distance from the sensor. This calibration is satisfactory as long as both air path contributions do not change. Regarding the reflection of the environment irradiance, the surrounding surfaces are usually much colder than the sensed surface; in that case, the reflection contribution is also negligible. For all these reasons, after a proper calibration of the optic instrument at each wavelength, one thus has access to the emitted radiance itself:
L(l,T) ¼ e(l)B(l,T) (6:11)
2. Airborne=Satellite Remote Sensing Transmission and air layer self-emission cannot be discarded anymore. Furthermore, the aimed surface is most often in the same temperature range as the environment whose emitted radiation is reflected on the surface (the ‘‘environment’’ consists of the atmosphere layer itself and nearby solid surfaces in the case of ‘‘rough’’ scenes like urban scenes). The complete equation involving Equations 6.7 and 6.6 has thus to be considered. Generally, however, the terrestrial surfaces are considered as Lambertian surfaces. After proper calibration, one has access to the spectral at-sensor radiance:
Ls(l,T, u,w) ¼ t(l, u,w)[e(l)B(l,T)þ (1 e(l))L#(l)]þ L"(l, u,w) (6:12)
In second one, the atmosphere contributions are so important that a supplementary task of atmospheric compensation needs to be accomplished.
