Satellite remote sensing is an important tool for investigating the atmosphere, climate and environment. It provides a global view which can be combined with regional measurements, for example at ground stations, to obtain an overall picture. Typical products resulting from remote sensing of the atmosphere are forecasts about the composition and characteristics of the atmosphere. Deriving geophysical quantities from remote sensing measurement data on the atmosphere for such purposes has two aspects. These relate to the fundamental functional relationship between the thermodynamic characteristics of the medium (such as temperature, pressure, density and composition) and the spectral optical characteristics (such as absorption and scattering) of the species involved (such as molecules and aerosols):
**Direct or forward problem:** efficient and precise modeling of radiative transfer in the atmosphere, whereby the goal is to determine the absorption or emission spectrum from given atmospheric characteristics
**Inverse problem:** finding numerically stable methods for estimating the desired geophysical values based on the measured spectra.

In **forward modeling** the requirements placed on the radiative transfer model are primarily determined by the measurement conditions, such as, especially, the spectral range, measurement geometry, and sensor type. Modeling a spectrum using the forward modeling approach can always only be regarded as an approximation, since many physical effects can merely be described in a simplified way in order to optimize the use of computer time. The main challenge in the ultraviolet and visible spectral range (UV/VIS) is, in particular, modeling multiple scattering as a radiation source. For instruments with high spectral resolution in the microwave and infrared range, by contrast, so-called line-by-line (LbL) radiative transport models are usually essential. Although in this case scatter can usually be ignored, this limits the relevance of the forward model to atmospheres which are almost free of clouds and aerosols.

Besides modeling actual atmospheric radiative transport, the forward model also has to be able to take into account modifications to the spectrum caused by the instrument, especially the sensor’s spectral and spatial influence. In addition, a precondition is complete familiarity with the spectroscopic molecule and aerosol data.

Inversion (or retrieval), in other words ascertaining the desired atmospheric parameters from the remote sensing measurement data, is usually accomplished with the help of numerical optimization methodologies. In practice this is accomplished by comparing the measurement data with simulated data generated by a suitable model of the measurement process (forward model) and an estimated parameter complement. Since the spectrum is usually a nonlinear function of the desired state variables, the analysis usually requires iterative refinement methods (least-squares). Beginning with an assumed atmospheric state vector, a spectrum is simulated with forward calculation and then compared with the measured spectrum. After suitable adjustment of the state vector, this procedure is repeated until the simulated and measured spectra sufficiently agree. Particularly in UV/VIS, differential methods without forward modeling have also become established (DOAS – Differential Optical Absorption Spectroscopy). Lately, neuronal networks are also being used, such as for processing data from the GOME sensor. Most inversion algorithms have to be supplied with a multitude of parameters. Besides selecting the measurement data (and in some cases also suitable spectral microwindows) and determining the quantities to be derived, this approach also requires the provision of additional parameters for the inversion program and the radiative transport program. And so, in the end, the process already begins when the inversion algorithm is designed, for example when mathematical routines are selected.

Precise dealing with radiative transfer must take into account the scattering of electromagnetic radiation caused by molecules and aerosols in the atmosphere. Looked at physically, it is a question of the interaction of a plane electromagnetic wave with a three-dimensional structure characterized by its geometry as well as by its dielectric properties. A theoretical description of this process must relate the wave’s degrees of freedom before and after the interaction so that measurements relevant for the scatter can be derived. In atmospheric remote sensing the scattering of electromagnetic waves caused by their impact with nonspherical particles is increasingly gaining in importance. Modeling this problem is considerably more complex than is the case of spherical scatter particles with application of the Mie theory. This is due both to the computational effort and to the significantly more complex convergence behavior of the available procedures. Scatter data banks for precisely defined nonspherical particles can be an aid here since they reduce the computational effort as well as free the user of remote sensing data from having to assess the underlying scatter models.