Optical imaging

In depth

Three kinds of light reach an optical sensor, drawn in Figure 1. Sunlight crosses the atmosphere, reaches the surface and is reflected towards the sensor. Some sunlight is scattered by the air into the sensor's view without ever reaching the ground, which is the path radiance. The sensor does not record reflectance but spectral radiance $L_\lambda$, power per unit area per unit solid angle per unit wavelength. A detector delivers a digital number $\mathrm{DN}$, and the calibration coefficients published for the instrument convert it back [1].

where $L_\lambda$ is the spectral radiance at the sensor in the band, in $\mathrm{W\,m^{-2}\,sr^{-1}\,\mu m^{-1}}$; $\mathrm{DN}$ is the digital number the detector records, a count without unit; $G$ is the gain of the band, in radiance units per count; and $O$ is its offset, in radiance units.

Here $G$ is the gain and $O$ the offset for the band. To compare scenes taken under different Sun angles and in different seasons, radiance is converted to top-of-atmosphere reflectance. With $d$ the Earth to Sun distance in astronomical units, $E_0$ the mean exoatmospheric solar irradiance in the band and $\theta_s$ the solar zenith angle,

where $\rho_{\mathrm{TOA}}$ is the top-of-atmosphere reflectance, without unit; $L_\lambda$ is the spectral radiance of equation 1, in $\mathrm{W\,m^{-2}\,sr^{-1}\,\mu m^{-1}}$; $d$ is the Earth to Sun distance, in astronomical units of $1.496\times 10^{11}\ \mathrm{m}$; $E_0$ is the mean exoatmospheric solar spectral irradiance in the band, in $\mathrm{W\,m^{-2}\,\mu m^{-1}}$; $\theta_s$ is the solar zenith angle; and $\pi$, in $\mathrm{sr}$, relates the radiance of a surface that scatters equally in all directions to the irradiance leaving it.

This follows from the definition of reflectance for a surface that scatters equally in all directions [1]. For a uniform surface of that kind, and ignoring multiple scattering between the ground and the air, the radiance at the sensor can be written as follows, where $\tau$ is the transmittance of the atmosphere along the path to the sensor, $\rho$ the surface reflectance, $E_g$ the irradiance reaching the ground and $L_p$ the path radiance [2].

where $L$ is the spectral radiance at the sensor and $L_p$ the path radiance, both in $\mathrm{W\,m^{-2}\,sr^{-1}\,\mu m^{-1}}$; $\tau$ is the transmittance of the atmosphere from the ground to the sensor, a fraction without unit; $\rho$ is the surface reflectance, without unit; $E_g$ is the spectral irradiance reaching the ground, in $\mathrm{W\,m^{-2}\,\mu m^{-1}}$; and $\pi$, in $\mathrm{sr}$, turns the irradiance reflected by a surface that scatters equally in all directions into radiance.

Equation 3 is the one that surface reflectance products invert, as the processing levels page explains. Scattering by air molecules is strongly dependent on wavelength, varying as $\lambda^{-4}$. Blue light at 450 nanometres is scattered about six times more than red at 700 nanometres, which is why distant scenes look hazy [3].