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Polarized Form of the Governing Equation Including Atmospheric Scattering Terms

In this chapter we will look at all the energy sources that contribute to the polarimetric radiance reaching the sensor. In Sec. 7.1 we introduce the radiometric terms of interest in the form of a governing equation. To better understand the polarimetric nature of these terms, we delve into the polarized behavior of atmospheric scattering in Sec. 7.2. The radiometric terms and concepts used throughout this section draw heavily on the radiometry fundamentals introduced in Chapter 2.

7.1 Governing Polarized Radiance Equation

In order to analyze remotely sensed polarimetric data, we need to develop a governing equation describing all the terms that contribute to the polarimetric radiance reaching the sensor. First, the radiometric equation for the unpolarized radiance is introduced and nomenclature is established. Then, the polarimetric representation of these equations is derived. This representation highlights the role of the polarimetric BRDF and guides a simplified field measurement technique introduced in Chapter 10. The polarized radiance governing equation forms the basis for all subsequent remote sensing studies. The approach presented here follows that of Shell and Schott (2005).

7.1.1. Scalar representation of the governing equation

The total radiance in the visible to near infrared (VNIR) portion of the spectrum (i.e., that of solar origin) reaching a sensor aperture (L s ) may be approximated as the sum of three radiance sources (see Schott (2007) for a more complete derivation of the terms in the governing equation):

(1) direct solar reflection from the target (L r )

(2) upwelled atmospheric radiance resulting from solar scatter along the target to sensor path (L u ) and

(3) target-reflected downwelled radiance from the skydome (L d ).