Light and the spectrum

In depth

Light is an electromagnetic wave. Its wavelength $\lambda$ and frequency $\nu$ are tied together by the speed of light $c$, and each photon carries an energy set by its frequency, where $h$ is the Planck constant.

where $c$ is the speed of light, $2.998\times 10^{8}\ \mathrm{m\,s^{-1}}$; $\lambda$ is the wavelength, in $\mathrm{m}$; and $\nu$ is the frequency, in $\mathrm{Hz}$.

where $E$ is the energy of one photon, in $\mathrm{J}$; $h$ is the Planck constant, $6.626\times 10^{-34}\ \mathrm{J\,s}$; $\nu$ is the frequency, in $\mathrm{Hz}$; $c$ is the speed of light; and $\lambda$ is the wavelength, in $\mathrm{m}$.

Short wavelengths therefore carry more energy per photon than long ones. Remote sensing names its regions of the spectrum by wavelength, as Figure 1 shows. The visible range runs from about 0.4 to 0.7 micrometres. Beyond the red lie the near infrared (reflected sunlight, used for vegetation), the shortwave infrared (reflected sunlight again, sensitive to moisture and minerals), and the mid-wave and thermal infrared (heat the surface emits itself). At wavelengths roughly a hundred thousand times longer lie the microwaves used by radar, which engineers label with letters such as X, C and L [1, 2].

Two sources of radiation dominate. The Sun, an object at about 5800 K, emits most strongly near 0.5 micrometres. The Earth's surface, at about 288 K, emits most strongly near 10 micrometres. Wien's displacement law gives both peaks from the temperature $T$ alone, with the constant $b = 2.898\times 10^{-3}\ \mathrm{m\,K}$.

where $\lambda_{\max}$ is the wavelength at which the spectral radiance of a blackbody peaks, in $\mathrm{m}$; $b$ is Wien's displacement constant, $2.898\times 10^{-3}\ \mathrm{m\,K}$; and $T$ is the absolute temperature of the body, in $\mathrm{K}$.

Substituting 5800 K gives 0.50 micrometres, and 288 K gives 10.1 micrometres. This is why a sensor in the visible and near infrared sees reflected sunlight, while one near 10 micrometres sees heat. The law is a consequence of the blackbody spectrum that Planck derived in 1901 [3], taken up again on the thermal imaging page.

The atmosphere then edits the signal. Water vapour, carbon dioxide and ozone absorb strongly in some ranges and hardly at all in others, leaving atmospheric windows. Optical and thermal sensors are built to look through these windows. Microwaves longer than a few centimetres pass through cloud and haze with little loss, which is why radar can image the surface in any weather [4, 2].

What a sensor ultimately wants is a property of the surface rather than of the sunlight. The spectral reflectance $\rho(\lambda)$ is the fraction of the light arriving at the surface that is sent back, with $\Phi_i$ the incident and $\Phi_r$ the reflected power at wavelength $\lambda$.

where $\rho(\lambda)$ is the spectral reflectance at wavelength $\lambda$, a fraction between 0 and 1 without unit; and $\Phi_r(\lambda)$ and $\Phi_i(\lambda)$ are the reflected and incident radiant power per unit wavelength, in $\mathrm{W\,m^{-1}}$.

Plotted against wavelength, reflectance is the spectral signature that later sections use to tell materials apart [4].