  

5.1 Radiative Transfer in the Earth’s Atmosphere An important tool to simulate changes in the solar radiation due to atmospheric scattering and absorption processes are radiative transfer models, also referred to as Forward Models. They provide the synthetic radiances, as they would be measured by the sensor, for a specified state of the atmosphere. These models are an important part of any retrieval process. Considering the entire atmosphere, one usually talks about the radiation field, which describes the angular and spatial distribution of the radiation in the atmosphere. The radiation field in the atmosphere is commonly characterized by its intensity which is defined as the flux of energy in a given direction per second per unit wavelength range per unit solid angle per unit area perpendicular to the given direction (Liou 2002). All interactions between the radiation and the atmosphere are classified either as extinction or as emission. The two processes are distinguished by the sign of the change of the radiation intensity as a result of the interaction. Extinction refers to any process which reduces the intensity in the direction under consideration and therefore includes absorption as well as scattering processes from the original direction into other directions. Emission refers to any process which increases the intensity in the direction under consideration and thus includes scattering into the beam from other directions, as well as thermal or other emission processes within the volume. In the description of radiative transfer presented here we neglect the polarization state of light for reasons of simplicity. However, we note that polarization is important for SCIAMACHY for two reasons, namely in order to 

 accurately simulate radiances in the UV and VIS   account for the polarization sensitivity of the instrument when determining the true radiance  

The general form of the radiative transfer equation describes all the processes which the radiance undergoes as a result of its interaction with a medium, taking energy conservation into account. It has the form (equ. 51) 



where I is the intensity of the radiation in a given direction, a is the extinction coefficient describing the fraction of the energy which is removed from the original beam and J is the source function which describes the energy emitted by the volume element, i.e., the increase of the intensity, I, in the original direction. If the energy of the radiance travelling in a certain direction through the atmosphere can only be increased due to the scattering processes – as is the case for the spectral range covered by SCIAMACHY – the source function depends on the intensity falling on the elementary volume from all directions (equ. 52) 



with being the scattering angle, i.e. the angle between the directions of the incident and scattered radiation, and is the single scattering albedo representing the probability that a photon which interacts with a volume element will be scattered rather than absorbed. The term denotes the probability that the radiation is scattered into a solid angle about a direction forming an angle with the direction of the incident radiation and the quantity is called the phase function. The total radiation field can be split into two components: the direct radiation, , which is never scattered in the atmosphere or reflected from the Earth's surface and the diffuse radiation, , which is scattered or reflected at least once. Since there is no relevant process in the atmosphere which increases the intensity of the direct solar radiation, the radiative transfer equation for the direct radiation leads to the homogeneous differential equation (equ. 53) 
with a solution described by the LambertBeer’s Law (equ. 54)
with being the solar irradiance and the integral along the photon path defining the optical depth . Integration is performed along the direct solar beam from the surface to the top of the atmosphere. This equation describes the attenuation of the direct solar or lunar light travelling through the atmosphere. Thus the direct component (equ. 54) simulates the radiative transfer when for example directly viewing the sun or the moon. Therefore, it describes SCIAMACHY measurements in occultation geometry. If the scattering processes in the atmosphere are nonnegligible ? as for SCIAMACHY nadir and limb measurements ? the diffuse component has to be considered in addition to the direct one. This leads to the standard radiative transfer equation for the diffuse component for a planeparallel atmosphere (equ. 55)
where denotes the scattering angle between the direct solar beam and the direction of observation.
Scattering occurs by molecules and various types of aerosols and clouds. Molecular scattering crosssections are characterised by the Rayleigh law, with aerosol scattering typically showing a much less pronounced dependence on wavelength (about ). Molecular scattering dominates in the UV with aerosols replacing molecules as the major source of scattered light in the VIS and NIR range (see fig. 53). The molecular scattering consists of two parts: the elastic Rayleigh component which accounts for 96% of scattering events and the 4% inelastic rotational Raman component, which is considered responsible for the so called Ring effect (?filling in? of solar Fraunhofer lines in the Earthshine spectra). Further details on atmospheric radiative transfer can be found in e.g. Liou (2002).(fig. 52)
fig. 52: 
The solar irradiance spectrum (red) and Earth radiance spectrum (blue) with a shape modified by absorption of trace gases and scattering in the atmosphere. (graphics: IUPIFE, University of Bremen) 

Direct and diffuse radiations are attributed to different light paths. Equations 54 and 55 need to be solved employing standard methods (Lenoble 1985). Idir and Idif must be added to describe the radiative transfer for the SCIAMACHY nadir geometry for not too large solar zenith angles (SZA). As an example, figure 52 shows the solar irradiance spectrum and the backscattered radiance at the top of the atmosphere. When SCIAMACHY nadir radiances at high SZA > 75° are simulated, the sphericity of the atmosphere must be taken into account in a pseudospherical approximation for the direct solar beam. For simulating SCIAMACHY limb radiances both the direct and the diffuse solar beam have to be treated in a spherical atmosphere, including refraction. The numerical solution of the radiative transfer equation is then accomplished by an iterative approach (Rozanov et al. 2001). Depending on the scientific application, several radiative transfer models are used in the SCIAMACHY data analysis. They include GOMETRAN (Rozanov et al. 1998), LIDORT (Spurr et al. 2001), DAK (Stammes 2001), SCIARAYS (Kaiser and Burrows 2003) and SCIATRAN 2.0 (Rozanov et al. 2005a).
These radiative transfer models are not only able simulating the radiance measured by SCIAMACHY, but they are also optimised to deliver additional parameters to quantify the geophysical parameters of interest. One of these parameters is the Weighting Function. This function is the derivative of the modelled radiance with respect to the model parameter, and describes how sensitive the radiance changes are when the parameters are modified. Another important quantity to be delivered by radiative transfer models is the Airmass Factor (AMF, see chapter 5.2) which is a measure for the photon path in the atmosphere.
fig. 53: 
Simulated vertical optical depth of the targeted constituents to be observed at 55? N around 10 a.m. The strong absorbers are plotted in the upper part and the relevant weak absorbers in the middle part. In the lower part the vertical optical depth due to Rayleigh scattering, aerosol extinction and absorption is given. Note the large dynamic range of the differential absorption structures used for retrieval of the constituents. (Graphics: IUPIFE, University of Bremen) 
