  

This results in (equ. 510) 



where subscripts n and n+1 denote the number of the iteration. The measurement error covariance matrix, Sy, is usually assumed to be diagonal, i.e. no correlation between measurement errors at different wavelengths or different tangent heights is considered. In the retrieval of the vertical distributions of the atmospheric species the a priori covariance matrix, Sa, is commonly chosen as a block diagonal matrix, i.e. vertical distributions of different atmospheric trace gases are assumed to be uncorrelated. The diagonal elements of Sa represent the variances of the vertical distribution of atmospheric trace gases, s, which e.g. can be derived from a climatology. The quality of the obtained solution is characterised by the a posteriori covariance matrix (equ. 511) 



and by the averaging kernels (equ. 512) 



characterising the response of the retrieved solution to the variation of the true atmospheric state. The root squares of the diagonal elements of the a posteriori covariance matrix are referenced to as the theoretical precisions. Employing the averaging kernels, the retrieved solution, x, can be related to the true solution, xtrue as (equ. 513) 



i.e., if the model state vector represents a vertical profile of an atmospheric trace gas, the retrieved values at each altitude are expressed as the sum of the a priori value at this altitude and of the deviation of the true profile from an a priori profile smoothed with the associated row of the averaging kernel matrix. For an ideal observing system, A is a unit matrix. In reality, the rows of the averaging kernel matrix are peaked with a finite width, which can be regarded as a measure of the vertical resolution of the retrieved profile. 5.4.2 Application of Inversion Theory to Limb Retrieval For the retrieval of the vertical distributions of atmospheric species from the measurements performed by SCIAMACHY the so called Global Fit technique is an effective way of implementing the inversion. The measurement vector contains the logarithms of the radiances in all selected spectral points and at all lineofsights, both for limb and occultation geometry, referenced to an appropriate irradiance spectrum. 
In the limb viewing geometry, the irradiance spectrum can be replaced by a limb measurement at an upper tangent height. Due to this normalisation the retrieval is relatively robust with respect to the radiometric calibration. In addition, the normalisation significantly reduces the sensitivity of the retrievals with respect to ground albedo and cloud cover. Commonly, before the main inversion step, a preprocessing is performed intended to correct for possible misalignment in the wavelength calibration and account for known atmospheric corrections such as the Ring effect. If required, a polynomial can be subtracted from all relevant spectra involved, accounting for both missing or inappropriate instrument calibration and unknown scattering properties of the atmosphere.
To illustrate the limb inversion in practice, fig. 512 shows an example of averaging kernels, weighting functions at 338.6 nm, and theoretical precision typical for BrO vertical profile retrieval from SCIAMACHY limb measurements. These results were obtained using the limb measurement at a tangent height of 38.5 km as a reference spectrum. As can be seen in the figure, the peak values of the averaging kernels are close to 1.0 only at tangent heights above 15 km. The peak value of about 0.55 at 12 km altitude indicates an increased dependence of the retrieved BrO amount at this altitude on BrO amount at neighbouring altitude levels and a priori information. Looking at the width of the averaging kernels, the height resolution of the measurements can be estimated to about 3 km, close to the geometrical resolution of the instrument. The weighting functions in the middle panel exhibit relatively sharp peaks near the tangent height down to 18 km tangent height, whereas at all lower tangent heights the weighting functions peak at about 18 km altitude. Nevertheless, the BrO amounts down to 12 km can be retrieved due to different shapes of the corresponding weighting functions. In accordance with the averaging kernels, the theoretical precision of the BrO vertical profile retrieval shown in the left panel has reasonable values of 1040% only above 14 km altitude and rapidly decreases below indicating the low information content in the measurements below 14 km. (fig. 512)
click to enlarge
fig. 512: 
Averaging kernels (left), weighting functions at 338.6 nm (middle), and theoretical precision (right) for BrO vertical profile retrievals from SCIAMACHY limb measurements. (graphics: IUPIFE, University of Bremen) 

To date, such type of retrieval algorithm and associated derivates has been used to obtain stratospheric profiles of O3, NO2 (Bracher et al. 2005, Sioris et al. 2004) and BrO (Rozanov et al. 2005b) from SCIAMACHY limb scattering profiles. This type of inversion algorithm was also applied and used to derive trace gas concentrations from lunar (Amekudzi et al. 2005a, 2005b) and solar occultation (Meyer et al. 2005) measurements. It is interesting to note that a similar approach as described above can also be applied to retrieve trace gas information from nadir measurements, as for example demonstrated for the Ozone profile retrieval from GOME nadir measurements (Munro et al. 1998, Hoogen et al. 1999).
All the applications listed above use a continuous spectral range to derive the trace gas information. Another group of limb inversion algorithms employs discrete spectral points in and outside strong trace gas absorption bands for the retrieval of vertical profiles. Retrieval algorithms utilising the difference in absorption between the centre and wings of the ozone Chappuis and Huggins bands were devised by Flittner et al. (2000). In a first step the limb radiance profiles are normalised with respect to a reference tangent height between 40 and 45 km. For the SCIAMACHY O3 Chappuis band retrieval (von Savigny et al. 2005a), the normalised limb radiance profiles are divided by the limb radiance profile at a nonabsorbing wavelength and then analysed in an optimal estimation scheme to retrieve the stratospheric O3 profiles. Retrievals in the O3 Chappuis band allow extraction of stratospheric O3 profiles for altitudes between about 1214 and 40 km.
  