Thermal imaging

In depth

Where the infrared bands sit

Infrared is a family of bands with different physics. The near infrared (0.7 to 1.4 micrometres) and the shortwave infrared (1.4 to 3 micrometres) carry reflected sunlight, and they are treated like the visible bands. The mid-wave infrared (3 to 5 micrometres) and the thermal infrared (8 to 14 micrometres) carry radiation that the surface emits itself. The two thermal ranges are atmospheric windows. The gap between them, from 5 to 8 micrometres, is largely closed by water vapour absorption [1, 2].

Emission and temperature

Every body above absolute zero emits radiation. A perfect emitter, a blackbody at temperature $T$, emits spectral radiance given by Planck's law [3], with the Boltzmann constant $k$.

where $B_\lambda(T)$ is the spectral radiance of a blackbody, in $\mathrm{W\,m^{-2}\,sr^{-1}\,m^{-1}}$ (divide by $10^{6}$ for the value per micrometre); $\lambda$ is the wavelength, in $\mathrm{m}$; $T$ is the absolute temperature, in $\mathrm{K}$; $h$ is the Planck constant, $6.626\times 10^{-34}\ \mathrm{J\,s}$; $c$ is the speed of light, $2.998\times 10^{8}\ \mathrm{m\,s^{-1}}$; and $k$ is the Boltzmann constant, $1.381\times 10^{-23}\ \mathrm{J\,K^{-1}}$.

Figure 1 plots equation 1 for three temperatures. Each curve is divided by its own peak so that only position and shape are compared, since the Sun's total emission is vastly greater than the Earth's. The peaks move to shorter wavelengths as the temperature rises, following equation 3 on the light and the spectrum page. A fire near 1000 K peaks near 3 micrometres, at the short end of the mid-wave band, which is why that band is used to find fires [2], and the Earth peaks within the 8 to 14 micrometre window. Integrated over all wavelengths, the emitted power per unit area rises with the fourth power of temperature, with the Stefan-Boltzmann constant $\sigma = 5.670\times 10^{-8}\ \mathrm{W\,m^{-2}\,K^{-4}}$.

where $M$ is the radiant exitance, the power emitted per unit area of surface over all wavelengths and directions, in $\mathrm{W\,m^{-2}}$; $\varepsilon$ is the emissivity of the surface, between 0 and 1 without unit and equal to 1 for a blackbody; $\sigma$ is the Stefan-Boltzmann constant, $5.670\times 10^{-8}\ \mathrm{W\,m^{-2}\,K^{-4}}$; and $T$ is the surface temperature, in $\mathrm{K}$.

Real surfaces are not blackbodies. Their emissivity $\varepsilon$, between 0 and 1, depends on material, moisture and roughness, and it multiplies the blackbody value. A sensor measures radiance $L$, and the simplest product inverts Planck's law for the temperature a blackbody would need to produce it, the brightness temperature $T_b$.

where $T_b$ is the brightness temperature, in $\mathrm{K}$; $L$ is the spectral radiance measured in the band, in $\mathrm{W\,m^{-2}\,sr^{-1}\,m^{-1}}$ like $B_\lambda$; $\lambda$ is the central wavelength of the band, in $\mathrm{m}$; and $h$, $c$ and $k$ are the constants of equation 1.

The brightness temperature is lower than the true surface temperature when $\varepsilon < 1$, and the atmosphere adds its own emission and absorption. Retrieving the land surface temperature therefore needs an estimate of emissivity and a correction for the atmosphere, done either with two neighbouring thermal bands (the split-window method) or with one band and atmospheric data. The methods and their accuracy are reviewed in the literature [4, 5, 6].