**Page updated:**
May 11, 2021 **Author:** Curtis Mobley

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# Problem Formulation

The introduction to atmospheric correction on the previous page illustrated the nature of the problem. It is now time to deﬁne the problem quantitatively.

The total radiance ${L}_{t}$ measured by a satellite-borne sensor at the top of the atmosphere (TOA) comes from contributions by atmospheric scattering, ${L}_{atm}$; Sun and sky radiance reﬂected back upward by the sea-surface and reaching the TOA, ${L}_{surf}^{TOA}$; and water-leaving radiance that reaches the TOA, ${L}_{w}^{TOA}$:

$${L}_{t}={L}_{atm}+{L}_{surf}^{TOA}+{L}_{w}^{TOA}\phantom{\rule{0.3em}{0ex}}.$$ | (1) |

For brevity, the viewing direction $\left({\mathit{\theta}}_{v},{\varphi}_{v}\right)$ and wavelength $\lambda $ are not shown. Expanding this equation into further levels of detail requires the deﬁnition of many diﬀerent radiances, and precise notation is needed to minimize confusion. The atmospheric contribution ${L}_{atm}$ is always considered to be at the TOA. However, the surface-reﬂected radiance and water-leaving radiance can be formulated either at the sea surface or at the TOA. For these radiances, a superscript TOA will be used to specify the TOA value. Thus ${L}_{w}$ will denote the water-leaving radiance just above the sea surface, and ${L}_{w}^{TOA}$ will denote how much of ${L}_{w}$ reaches the TOA. Table 1 summarizes the various radiances introduced in this chapter and used throughout this report.

Symbol | Deﬁnition |

${L}_{t}$ | total upwelling radiance at the top of the atmosphere |

${L}_{atm}$ | total contribution of atmospheric scattering to the TOA radiance |

${L}_{surf}^{TOA}$ | total contribution of surface-reﬂected radiance to the TOA radiance |

${L}_{R}$ | total Rayleigh radiance at the TOA |

${L}_{r}$ | ”standardardized” Rayleigh radiance at the TOA |

${L}_{a}$ | TOA radiance due to scattering by aerosols only |

${L}_{aR}$ | TOA radiance due to aerosol-molecule scattering |

${L}_{A}$ | ${L}_{a}+{L}_{aR}$; total aerosol radiance at the TOA |

${L}_{w}$ | water-leaving radiance just above the sea surface |

${L}_{w}^{TOA}$ | the part of the water-leaving radiance ${L}_{w}$ that reaches the TOA |

${L}_{g}$ | direct Sun glint radiance just above the sea surface |

${L}_{g}^{TOA}$ | the part of the direct Sun glint radiance ${L}_{g}$ that reaches the TOA |

${L}_{sky}$ | surface-reﬂected background sky radiance at the sea surface |

${L}_{sky}^{TOA}$ | the part of the surface-reﬂected background sky radiance ${L}_{sky}$ that reaches the TOA |

${L}_{wc}$ | radiance due to whitecaps and foam just above the sea surface |

${L}_{wc}^{TOA}$ | the part of the whitecap radiance ${L}_{wc}$ that reaches the TOA |

${L}_{u}$ | upwelling underwater radiance just beneath the sea surface |

The atmospheric contribution in Eq. (1), usually called atmospheric path radiance, comes from scattering by atmospheric gases and aerosols, including multiple scattering between gases and aerosols. The path radiance that comes solely from scattering by atmospheric gas molecules is usually called the Rayleigh radiance, ${L}_{R}$, because scattering by molecules is well described by the Rayleigh mathematical model of scattering by particles that are much smaller than the wavelength of light. In the absence of any aerosols, the atmospheric path radiance would equal the Rayleigh radiance. Let ${L}_{a}$ denote the aerosol contribution, which is the path radiance that would occur if the atmosphere consisted only of aerosol particles. Let ${L}_{aR}$ denote the contribution resulting from multiple scattering between aerosols and gases. The total surface reﬂectance can be separated into a contribution due to direct Sun glint from the water surface, ${L}_{g}^{TOA}$; by background sky radiance reﬂected by the water surface, ${L}_{sky}^{TOA}$; and by Sun and sky radiance reﬂected by whitecaps and foam, ${L}_{wc}^{TOA}$. Thus Eq. (1) can be further partitioned into

$${L}_{t}={L}_{R}+\left[{L}_{a}+{L}_{aR}\right]+{L}_{g}^{TOA}+{L}_{sky}^{TOA}+{L}_{wc}^{TOA}+{L}_{w}^{TOA}\phantom{\rule{0.3em}{0ex}}.$$ | (2) |

In practice, the aerosol and aerosol-gas contributions are usually grouped together and treated as one contribution, sometimes denoted ${L}_{A}={L}_{a}+{L}_{aR}$ and often called just the aerosol contribution. The sky reﬂectance term is accounted for as part of the Rayleigh correction, which incorporates reﬂectance by the sea surface. For some sensors that were speciﬁcally optimized for ocean color (e.g., CZCS and SeaWiFS), the strongest part of the Sun glint (the Sun’s glitter pattern) is avoided by pointing the sensor in a direction away from the Sun so that almost no direct glint is present in the image. However, there is still a correction for residual amounts of Sun glint. Figure 1 illustrates these contributions to the TOA radiance.

Most papers (e.g., Wang and Bailey (2001), Wang (2002)) rewrite Eq. (2) as

$${L}_{t}={L}_{R}+\left[{L}_{a}+{L}_{Ra}\right]+T{L}_{g}+t{L}_{wc}+t{L}_{w}\phantom{\rule{0.3em}{0ex}},$$ | (3) |

or something very similar. Now, however, ${L}_{g}$, ${L}_{wc}$, and ${L}_{w}$ are all measured at sea level. $T$ is the direct transmittance between the sea surface and the TOA along the viewing direction, and $t$ is diﬀuse transmittance in the viewing direction. These transmittances are discussed on the Atmospheric Transmittances page.

Yet a third formulation can be found in the literature (e.g., Franz et al. (2007), Eq. 1):

$${L}_{t}=\left(\right.{L}_{r}+\left[{L}_{a}+{L}_{ra}\right]+{t}_{dv}{L}_{wc}+{t}_{dv}{L}_{w}\left)\right.\phantom{\rule{0.3em}{0ex}}{t}_{gv}{t}_{gs}{f}_{p}\phantom{\rule{0.3em}{0ex}}.$$ | (4) |

Here ${t}_{dv}$ is the diﬀuse transmittance along the viewing path of the sensor. ${t}_{gv}$ is the transmittance by atmospheric gases in the viewing direction, and ${t}_{gs}$ is the transmittance by atmospheric gases in the Sun’s direction; these transmittances are usually called gaseous transmittances. ${f}_{p}$ is a known instrument polarization-correction factor. Comparison of Eqs. (3) and (4) shows, for example, that

$${L}_{R}={L}_{r}{t}_{gv}{t}_{gs}{f}_{p}\phantom{\rule{0.3em}{0ex}}.$$ |

Thus the total TOA Rayleigh contribution ${L}_{R}$ has been factored into a product of terms involving a Rayleigh term times gaseous transmittances and a polarization-correction factor. The diﬀerence between Eqs. (3) and (4) is primarily a matter of simpliﬁcation for presentation purposes. The ${f}_{p}$ term came into the nomenclature because MODIS has large polarization sensitivity and this requires correction. Earlier papers by Gordon and Wang often ignored the gaseous transmission terms because they were only considering ozone, which could be “taken oﬀ the top,” so to speak, with the remaining problem being eﬀectively formulated below the ozone layer. The ${L}_{r}$ term is computed using a standard atmosphere and only non-absorbing gases ${N}_{2}$ and ${O}_{2}$. This allows ”standard” Rayleigh radiances ${L}_{r}$ to be computed as a function of Sun and viewing geometry. The gaseous transmittances are computed by use of absorption coeﬃcients, computed path lengths, and gas concentrations for the various gases. The ${f}_{p}$ term is computed for each image pixel as a function of atmosphere and surface polarization states (modeled Rayleigh and glint Stokes vectors) and the sensor-speciﬁc polarization sensitivity with viewing direction.

All of Eqs. (2), (3), and (4) can be found in the literature. They all give the same TOA total radiance ${L}_{t}$. Which form is used in a particular instance is determined by convenience. Forms (2), (3) are often convenient for discussions of theory, whereas form (4) is convenient for operational atmospheric correction algorithms.

The goal of atmospheric correction is to convert a measured top-of-the-atmosphere radiance ${L}_{t}$ into the corresponding sea-level water-leaving radiance ${L}_{w}$. Since only ${L}_{t}$ is measured, this requires estimation of the various atmospheric and surface-reﬂectance terms seen in Eqs. (3) or (4) so that they can be subtracted from ${L}_{t}$ in order to arrive at ${L}_{w}$. How this is done is discussed in Level 2 of this Chapter.