2.1. Optimal Estimation Approach (Five Algorithms)
 Ozone profile retrieval from UV reflectance measurements is an ill-posed problem which can only be solved by applying suitable constraints. The well-known nonlinear OE approach outlined by Rodgers  iteratively applies the linear OE formula to find the cost function minimum. This method can be used for the inversion of weakly nonlinear forward models, and solves ill-posed problems by using a priori information as regularization constraint. In the retrieval the spectral measurement is related to an atmospheric profile with a forward model; the use of an a priori profile and its covariance matrix stabilizes this inversion by providing suitable constraints for a solution.
 The retrieved ozone profile from the optimal estimation method can be regarded as a weighted average between a priori and measurement information. This is reflected in [Rodgers, 2000, p. 31]
with the matrix A having elements of
and for the OE methods A is given as [Rodgers, 2000, p. 67]:
 In these equations, xretrieved, xa priori, and xtrue are vectors of ozone number densities at the altitude levels zi=1,n of the retrieval algorithm and they correspond to the values of the retrieved, a priori and true (i.e., observed) state, respectively. A is the so-called averaging kernel matrix, or model resolution matrix. Reorganization of equation (1) shows that A characterizes the mapping between (1) the difference between the true and the a priori profile (true anomaly) and (2) the difference between the retrieved and a priori profile (retrieved anomaly). In equation (3), Sa and S are the a priori and measurement error covariance matrices, respectively, and K is the so-called weighting function matrix, which describes how the forward model (F(x)), that relates the spectral measurement to the true state vector, is sensitive to changes in the state vector, i.e., Kij = ∂Fi(x)/∂xj.
 In case of ozone profile retrieval from UV spectra of nadir-viewing instruments like GOME, the averaging kernel reflects the limited sensitivity of the spectral measurement to fine-scale structures and to the profile below the ozone maximum. In addition, the kernels are dependent on the detailed specification of the state vector, a priori and measurement errors which are particular to a specific retrieval scheme. For example, as evident from equations (1) and (3), for smaller measurement errors the averaging kernel matrix tends toward the identity matrix and hence the OE solution becomes less dependent on the a priori profile. For larger measurement errors the averaging kernel elements go to zero and the solution relies more on the a priori. For the a priori errors the situation is reverse. Therefore the “choice” for the settings of the input measurement and a priori errors is important in the design of a retrieval system. Although there is a consensus for the measurement error, there is no such thing for the a priori. The complete a priori covariance matrix is generally constructed assuming an exponential decrease from the diagonal value (i.e., a priori variance) using a correlation length. The off-diagonal elements of Sa can then be written as
 with b = 1 or , depending on whether Rodgers [1990, equation 16] or Hoogen et al. [1999, equation (3)] is followed, respectively, for the chosen functional decay. The more determining factor in equation (4) is the choice for l, which is the so-called correlation length. The distance at which the covariance has decreased by e−1, from the variance at the nominal altitude, can be directly compared between schemes that use equation (4) with different values for b. This distance is reached when (z–z′) equals l and then allows direct comparison of l.
 Equation (1) reflects the deviation between the true and the retrieved profile, which is especially important when comparing the retrieved profile to correlative measurements. In the two extreme cases when (1) A is the identity matrix: the retrieved and the true profiles are equal, and (2) all elements of A are zero: the retrieved profile equals the a priori values. A detailed analysis of the averaging kernels is presented in section 3.
2.1.1. IUP Algorithm
 The Institute of Environmental Physics at the University of Bremen in Germany (IUP) has developed the full retrieval method (FURM) algorithm, and data presented here stem from version 5.0. This algorithm is based on the OE method but with the addition that it includes the information matrix method from Kozlov , which adapts the number of fit parameters to the measurement information content [Hoogen et al., 1999].
 The radiative transfer model (RTM) GOMETRAN, specifically designed for GOME retrieval applications [Rozanov et al., 1997, 1998], is used for calculating radiances and weighting functions. Besides ozone eigenvectors, other atmospheric parameters such as aerosol, temperature, NO2, albedo, Rayleigh scattering and the so-called Ring effect are simultaneously fitted. These parameters show negligible correlation among each other. After each iteration step a shift and squeeze between the wavelength axes of radiance, irradiance and cross sections is performed for wavelength adjustments. The Ring effect, or the filling in of solar absorption lines, can be explained by rotational Raman scattering and is taken into account by using look-up tables, for various atmospheric scenarios and solar zenith angles, to correct the GDP spectra.
 Clouds are treated as highly reflecting surfaces at 0-km altitude (clouds as albedo approach), which means that in the RTM the spherical albedo represents the weighted mean of surface and cloud albedo, the weight being the fractional cloud cover. Initially, albedo (and hence cloud) information is derived from the PMDs, and then it is further adjusted as part of the fitting process. In the RTM the Earth's surface is assumed to be a Lambertian reflector with wavelength-dependent albedo.
 The GDP spectra contain unresolved problems with the radiometric calibration, particularly between 260 and 290 nm [Hilsenrath et al., 1996]. In the retrieval they appear as spectral fit residuals with characteristic structures, but none of the atmospheric fit parameters can account for them [Hoogen et al., 1999]. An empirical calibration correction function was derived separately for bands 1 and 2. In the retrieval algorithm third-order Chebyshev polynomials are taken into account as additional fitting parameters using the coefficients of the correction function. Below 290 nm, there are also strong NOγ spectral features in the measurements [McPeters, 1989], which cannot appropriately be taken into account in the RTM. These two reasons lead to the restriction in the IUP algorithm (version 5.0) of only fitting wavelengths longer than 290 nm. In a more recent development, a new calibration correction scheme has been introduced that permits adding the wavelength range 275–290 nm to the fit window [Tellmann et al., 2004], but this new version (6.0) was not available for this comparison.
 The a priori ozone profiles used in the IUP algorithm are from the global ozone climatology of Fortuin and Kelder , which is based on ozonesonde and satellite measurements. This climatology provides monthly zonal mean ozone profiles in 10° latitude bands. The a priori variance of these profiles is fixed to 30%. The a priori covariance matrix is generated following equation (4) with b = and using a correlation length of 5 km. The temperature profiles are taken from the UK Met Office (UKMO) analysis [Swinbank and O'Neill, 1994], and are used to take into account the temperature dependence of the ozone cross sections.
2.1.2. KNMI Algorithm
 The Royal Netherlands Meteorological Institute (KNMI) developed the ozone profile retrieval algorithm (OPERA), and version 1.3 was used to generate data for this paper. Note that this algorithm is different from the retrieval algorithm described in the paper of van der A et al. , and the main difference is the RTM used. Ozone profiles are derived from the GOME data in the wavelength range 270–330 nm, and the spectra are coadded for the data coming from band 1b and 2. The radiometric and wavelength calibration of the GDP level 1 data are too inaccurate for ozone profile retrieval, and therefore several corrections are applied using the spectral calibration program GOMECAL (available through http://www.knmi.nl/gome_fd/gomecal/). This involves an improved wavelength calibration [van Geffen and van Oss, 2003], an improved correction for the polarization sensitivity of GOME [Schutgens and Stammes, 2003] and a radiometric recalibration involving a time-independent and a time-dependent (degradation) correction [van der A et al., 2002].
 The Sun-normalized radiances are simulated by constructing an atmospheric model and running the Linearized Discrete Ordinate Radiative Transfer model LIDORTA in six streams [van Oss and Spurr, 2002]; the number of streams sets the angular resolution of the model. LIDORTA is a simplified and sped-up version of the full LIDORT model [Spurr et al., 2001] replacing several numerical solvers with analytical solutions. LIDORTA is only applied for the multiple scattered part of the radiance and runs with a limited set of 20 layers. The single scattering part is computed with a dedicated, simpler and therefore faster, single scattering model with the full retrieval grid of 40 layers. In the model, the ozone profile elements that are actually fitted for are layer column amounts at a fixed vertical grid. In this version of the algorithm, the layers are chosen in such a way that they have the required GOME1-O3P-WG altitude levels at their centers.
 LIDORTA treats the sphericity of the atmosphere both for the solar direct beam and the line of sight by a pseudospherical approximation. LIDORTA is a scalar model in the sense that it does not treat polarization and the vector nature of the radiation field. This gives errors for the radiance at the top of the atmosphere in the wavelength range used that can reach 10% for scattering angles of 90°. This error is corrected for using a precomputed look-up table containing the scalar error for the complete range of wavelengths, atmospheric and viewing conditions. Raman scattering (responsible for the Ring effect) is not treated in the RTM, but accounted for using a high-resolution spectrum convolved with the Raman lines [Chance and Spurr, 1997], and the amplitude for this Ring spectrum is fitted as an auxiliary parameter.
 The atmospheric model used in the KNMI algorithm treats (fractional) cloud cover as a Lambertian reflecting layer at the cloud top height for the fraction of the pixel covered with clouds. The effective cloud fraction and cloud top height are obtained from the Fast Retrieval Scheme for Cloud Observables (FRESCO) extracting information from the oxygen A band [Koelemeijer et al., 2001]. By fitting an effective cloud fraction, the presence of aerosols is partly taken into account in the FRESCO retrieval. The error made with this procedure is smaller than when taking a (random) guess at the unknown aerosol distribution (confirmed by Boersma et al.  for GOME NO2 retrievals). The surface albedo is fitted for cloud fractions <0.15, and for all other cases the albedo of the cloud.
 For the ozone cross sections, OPERA uses the temperature-parameterized data set of Bass and Paur  and Paur and Bass , corrected according to Orphal . Trace gases other than ozone are not treated and assumed not to affect the retrieval in this spectral range. The a priori ozone profile information comes from the global ozone climatology of Fortuin and Kelder , with covariance information derived from the same data set [Bhartia, 2002], which corresponds to a correlation length of 4–5 km in equation (4) with b = .
2.1.3. RAL Algorithm
 The Rutherford Appleton Laboratory (RAL) has developed a three-step scheme to retrieve ozone profiles spanning troposphere and stratosphere [Siddans, 2002; Munro et al., 1998]. Version 2.0 of the retrieval scheme was used for this paper. In step 1, an ozone profile is retrieved from Sun-normalized radiances at selected wavelengths of the ozone Hartley band (GOME band 1) in the range 265–307 nm. Information from this spectral range is primarily on stratospheric ozone. A priori ozone profile comes from the Fortuin and Kelder  climatology except that in the troposphere a fixed value of 1018 molecules/m3 is assumed (1.5–2 times larger than the climatological values). The a priori uncertainty is set by default to 100% for retrieval levels at 0, 6 and 12 km, 30% at 16km, 10% from 20–52 km, 50% at 56km, and 100% from 60–80km (retrieval levels are spaced with 4-km intervals throughout the stratosphere and mesosphere). The default uncertainty is replaced by the Fortuin and Kelder  climatological relative variability at altitudes where the latter exceeds the former. A vertical correlation length of 6 km is applied to generate the covariance matrix using equation (4) with b = 1. The surface albedo, a scaling factor for the Ring effect and the dark signal are retrieved jointly.
 In step 2, the surface albedo for each of the eight band 2 ground pixels is retrieved from the Sun-normalized radiance spectrum between 335 and 340 nm. Then, in step 3, information on lower stratospheric and tropospheric ozone is added by exploiting the temperature dependence of the spectral structure in the ozone Huggins bands. The wavelength range 323–334 nm (GOME band 2) is used in conjunction with UKMO analyzed temperature profiles [Swinbank and O'Neill, 1994]. Each direct Sun band 2 spectrum is fitted to a high-resolution (0.01 nm) solar reference spectrum to improve knowledge of wavelength registration and slit function width.
 In the Huggins bands fit, the log of Sun-normalized radiance is taken and a low (third) order polynomial is subtracted, allowing differential structures to be fitted to a precision of <0.1% root-mean-square (compare ∼1% in the Hartley band). This differential approach in step 3 leads to improvements in the tropospheric retrieval and results in less stringent requirements on the absolute radiometric accuracy. In this step the a priori ozone profile and its error are the output from step 1, except that an a priori correlation length of 8 km is imposed.
 The RTM is derived from GOMETRAN [Rozanov et al., 1997], but the original code has been modified substantially in order to increase its efficiency without losing accuracy. Within the RTM there is no explicit representation of clouds, which are treated as highly reflecting surfaces at 0-km altitude (clouds as albedo approach, see section 2.1.1). When clouds are present, a negative bias in retrieved ozone below the actual cloud top height is therefore to be expected from this scheme.
2.1.4. SAO Algorithm
 The Smithsonian Astrophysical Observatory (SAO), a research institute of the Smithsonian Institution, is a part of the Harvard-Smithsonian Center for Astrophysics (CfA). The SAO algorithm, version 0.9, also uses the OE approach to derive ozone profile information. This algorithm performs a detailed treatment of (1) variable slit width in the instrument transfer function, (2) variable wavelength shift between radiances, irradiances and spectroscopic data, (3) real-time first-order Ring effect correction [Sioris and Evans, 2000], (4) undersampling correction [Chance et al., 2005], and (5) polarization correction.
 Ozone profiles are retrieved from GDP data with the GOMECAL polarization correction only (i.e., not using the other GOMECAL correction options). To reduce measurement errors and because of relatively broad ozone absorption structure in 289–307 nm, five neighboring pixels (i.e., in wavelength grid) are coadded and sampled at every 2 pixels. LIDORT [Spurr et al., 2001] is used to simulate radiances and weighting functions with similar polarization correction to the KNMI algorithm. The state vector includes ozone number density at 26 levels of 2 km from 0− to 50−km altitude, surface albedo, scaling parameters for Ring effect and undersampling correction, and scaling and shift parameters for other trace gases (NO2, SO2, BrO).
 In the characterization of the atmosphere, the SAO algorithm uses monthly mean stratospheric aerosol data from SAGE-II [Bauman et al., 2003] and tropospheric aerosol model fields from the Global Ozone Chemistry Aerosol Radiation and Transport (GOCART) model [Chin et al., 2002] as described by Martin et al. . Clouds are treated as Lambertian surfaces, and cloud fraction and cloud top pressure come from the GOME cloud retrieval algorithm (GOMECAT, which was formerly abbreviated as CRAG) [Kurosu et al., 1999]. An initial surface albedo is derived from the spectral measurements at 370 nm, where atmospheric absorption is minimal. The SAO algorithm uses daily European Centre for Medium-Range Weather Forecast (ECMWF) temperature profiles and National Centers for Environmental Prediction/National Center for Atmospheric Research (NCEP/NCAR) surface pressure. The Total Ozone Mapping Spectrometer (TOMS) version 8 ozone profile climatology [Bhartia and Wellemeyer, 2002] with Earth-Probe TOMS monthly mean total ozone is used to initialize a priori ozone profiles. This climatology has 3–10 profiles for each 10° latitude band and month, and the Earth-Probe TOMS monthly mean total ozone is used to select the appropriate a priori ozone profile from this data set. The a priori standard deviations are based on a 15 year ozone profile climatology, from SAGE and ozonesonde data [McPeters et al., 2003], with the following changes: the standard deviations at 40–50 km are assumed to be 70, 60, 50, 40, 30, 20% for 50, 48, 46, 44, 42, 40 km, respectively; and between 0 and 10 km, the a priori standard deviations are required to be at least 40%. The off-diagonal elements of the covariance matrix are generated using equation (4) with b = 1 and a correlation length of 6 km. In addition, the ozone profiles above 50 km are fixed using climatological values [Bhartia and Wellemeyer, 2002].
2.1.5. NOAA Algorithm
 The National Oceanic and Atmospheric Administration (NOAA) has applied the version 8 SBUV/2 algorithm, developed for the SBUV instruments, to the GOME data. Version 8 was derived from the version 6 algorithm which is described by Bhartia et al. . Unlike the previous four OE algorithms, this algorithm was not specifically designed for GOME data. The SBUV data are measured at the following wavelengths (nm): 251.99, 273.51, 283.27, 287.62, 292.26, 297.54, 301.93, 305.80, 312.50, 317.51, 331.23, and 339.84 with a bandwidth of ∼1.1 nm. A triangular filter centered at those values has been used to convert GOME spectral data to the SBUV band pass. Because the GOME data have large errors below 270 nm, an extrapolation was used to provide the standard input for the SBUV retrieval algorithm at 251.99 nm.
 The SBUV algorithm uses a single-scattering forward model calculation coupled with adjustments from multiple scattering tables created from the RTM developed for TOMS, called TOMRAD. This model is based on successive iteration of the auxiliary equation in the theory of radiative transfer developed by Dave . This solution accounts for all orders of scattering, as well as the effects of polarization, by considering the full Stokes vector in obtaining the solution. However, the solution is limited to Rayleigh scattering only and can only handle reflection by Lambertian surfaces. Modifications that have been incorporated into the code include a pseudospherical correction, molecular anisotropy [Ahmad and Bhartia, 1995], and rotational Raman scattering [Joiner et al., 1995]. In the pseudospherical correction, the incoming and the outgoing radiation is corrected for changing solar and satellite zenith angle due to Earth's sphericity but the multiple scattering takes place in plane parallel atmosphere. Comparison with a full-spherical code indicates that this correction is accurate to 88° solar zenith angle [Caudill et al., 1997]. For the cloud calculations, the algorithm uses an 1° × 1° climatology of monthly cloud top pressures [McPeters et al., 2003], and a similar snow/ice climatology. If snow or ice is present the clouds are treated as though they are at the surface.
 The version 8 SBUV(/2) algorithm has its own a priori ozone [McPeters et al., 2003] and temperature profile database, but they were not used for the retrieved data in this paper. For this study, the NOAA algorithm used the a priori profiles supplied by KNMI (i.e., from Fortuin and Kelder  and ECMWF analysis for ozone and temperature, respectively). The a priori covariance is constructed as follows: the diagonal elements correspond to 50% variance and the nondiagonal covariance elements fall off with a correlation length of approximately two Umkehr layers (∼10 km), using equation (4) with b = . The measurement covariance is diagonal and corresponds to radiance errors of 1% in each band.
2.2. Phillips-Tikhonov Regularization Approach (One Algorithm)
2.2.1. General Description
 The PTR approach [Phillips, 1962; Tikhonov, 1963] has been little used for the analysis of atmospheric spectra, e.g., to retrieve ozone profiles. However, it has been extensively studied in the mathematical field of inversion. The analysis of the fundamental problem by Hansen and O'Leary  and Hansen  provides a basis for the application of PTR to remote sensing problems. In contrast to the OE approach, the PTR approach does not require a priori ozone profiles and corresponding covariance matrices, but uses a smoothness constraint to determine the amount of information that can be retrieved from the measurement. Nevertheless, the same equations (1) and (2) are applicable, but here the vector xa priori is zero.
2.2.2. SRON Algorithm
 The inversion model of the algorithm developed by the Space Research Organization Netherlands (SRON) treats the ill-posed problem of ozone profile retrieval using the PTR approach [Hasekamp and Landgraf, 2001]. In addition to the least squares minimization between forward model and measurement, this algorithm includes minimization of the first derivative norm of the profile as a side constraint. The minimization of the least squares condition and the minimization of the first derivative norm are balanced by a regularization parameter. The rationale behind the minimization of the first derivative norm as a side constraint is that the measurement is insensitive to fine-scale structures of the ozone profile. These vertical structures do not influence the residual norm but strongly influence the first derivative norm. The regularization parameter should be chosen such that the retrieved profile contains vertical structures that most significantly influence the measurement, while the structures to which the measurement is insensitive should be filtered out. Such a value of the regularization parameter is found from the L curve [Hansen and O'Leary, 1993], which is a parametric plot of the first derivative norm versus the least squares norm that has an L-shaped corner. The regularization parameter corresponding to the corner of the L curve yields a good balance between the two minimizations.
 The forward model of the SRON algorithm consists of a RTM based on the Gauss-Seidel iteration technique, which fully includes multiple scattering and polarization. This model simultaneously calculates the four Stokes parameters at the top of the atmosphere and the corresponding analytical weighting functions, which are essential for any physically based retrieval algorithm. The RTM is described in detail by Landgraf et al.  for the scalar case and the extension to polarization is described by Hasekamp and Landgraf [2002a]. The inclusion of polarization in the radiative transfer calculations overcomes errors of up to 10% made by the commonly used scalar RTMs, which generally neglect the polarization properties of light. Another advantage of this RTM is that it allows a direct modeling of the polarization-sensitive GOME measurement using the Mueller matrix formalism. Therefore the SRON algorithm can be directly applied to the GOME measurements, which are thus not corrected for polarization (see Table 1). In this way the SRON algorithm is independent of the (optional) polarization correction of the GOME data processor, which can cause errors in the GOME spectra of up to 8% [Hasekamp et al., 2002].
 The additional fit parameters included in the SRON algorithm are a Lambertian surface albedo, a wavelength shift to correct for calibration errors, and the amplitude of a Ring spectrum precalculated by the code of Landgraf et al. . The effect of clouds is accounted for by using the independent pixel approximation, which separates the radiative transfer calculations for the cloudy and the cloudless scenes, with cloud fractions and cloud top heights from FRESCO. The ozone cross sections used in the SRON algorithm are those described by Voigt et al. .
2.3. Neural Network Approach (Two Algorithms)
 The NN approach uses a fully feed forward neural network, also called multilayer perceptron (MLP) [Rumelhart et al., 1986], which can be applied to generate a mapping between GOME spectral data, other supplementary input parameters and the output ozone distribution. A training data set is used to derive the mapping between various input parameters and the known collocated ozone distributions. Unlike the other retrieval schemes, which use a physical approach, this approach uses all available information in a primarily statistical way. One of the main advantages of the NN is that once it is trained, which is a slow process, it is several orders of magnitude faster than the other approaches. The main disadvantage, or restriction, is the reliance on the training data set, which should be large in volume and of the highest available quality, in terms of accuracy, precision and vertical resolution. The data quality of the NN output can never be better than the quality of the training data.
2.3.1. UTV Algorithm
 Tor Vergata University (UTV) has developed a NN scheme to derive ozone profiles from GOME spectra. The underlying idea of the algorithm is to train a NN using already existing RAL retrieved ozone profiles [Munro et al., 1998] and to use the trained net for new estimations [Del Frate et al., 2002]. The method takes advantage of both the high retrieval accuracy characterizing the profiles provided by RAL, and of the potentialities of the NN which after the training process is able to give new estimations in real time. Although there are better training data sets available and the data quality of the UTV algorithm will never be better than the RAL data, the advantage of using these data is its large volume for training and perfect match in collocation. It should be noted that the RAL products used in the construction of the UTV algorithm are different (older version) from those presented by RAL in this paper.
 The GOME data used in the scheme consist of solar irradiance and Earth radiance spectra from GDP. The solar irradiance spectra are measured daily by GOME and are used as the reference light source spectra. The selected wavelength range is 321–325 nm with a spectral resolution of 0.12 nm, which is based on a spectral calibration performed using a fourth-order polynomial and has been chosen according to four requirements. First, in this range, there is a higher spatial resolution, with respect to the Hartley ozone absorption band, due to a shorter integration time. Second, this range is characterized by a high value of the signal-to-noise ratio. Third, in this range, there is a high-temperature dependence of the ozone cross sections [Burrows et al., 1999b]. Fourth, in this range, there is the possibility to compute the ozone slant path content using the Temperature Independent Differential Absorption Spectroscopy (TIDAS) method [Zehner and Casadio, 2000].
 The Earth radiance spectra also undergo a normalization procedure in order to eliminate as much as possible the effects of instrumental parameters on the spectral shape. As far as the topology of the NN is concerned, a MLP-type network with one hidden layer is considered. The input vector consisted of the 26 selected GOME channels plus the solar zenith angle and the ozone slant path, and also the hidden layer has 28 units. Minimization of the error function has been pursued by a scaled conjugate gradient (SCG) algorithm [Müller, 1993].
2.3.2. ZSW Algorithm
 The Center for Solar Energy and Hydrogen Research (ZSW) in Stuttgart, Germany, has developed a NN scheme called Neural Network Ozone Retrieval System (NNORSY), and version 1.2 was used to generate data for this paper. In contrast to the UTV approach, the nonlinear regression performed in the ZSW algorithm infers the vertical distribution of ozone from a combination of climatological (latitude, season), meteorological (temperature) and spectral information (GOME spectra, solar zenith angle, scan angle, sensor age) [Müller et al., 2003; Müller, 2002]. The system effectively learns to correlate the behavior of atmosphere and sensor, even as the sensor characteristics slowly change over time due to, e.g., degradation. Thus only the basic GDP calibration procedure for level 1 data is performed. ZSW algorithm employs about 100 GOME spectral values covering the wavelength ranges 290–325 nm (Hartley/Huggins band), 380–385 nm (atmospheric window), 598–603 nm (Chappuis band), and 758–772 nm (oxygen A band) [Müller et al., 2003]. Employing additional cloud or ground albedo information was found to be unnecessary.
 UKMO analyzed temperature profiles [Swinbank and O'Neill, 1994] were included as a predictor, since the stratospheric part of the atmospheric temperature field is known to correlate strongly with ozone. In a NN the effect of using temperature information is quite different from the usage in an RTM, where only its comparatively small effect on ozone absorption can be exploited.
 In the ZSW algorithm the MLP is trained by means of a modified Resilient Propagation algorithm [Riedmiller, 1994], which is a fast heuristic approximation for a second-order function minimization scheme [Bishop, 1995]. Knowledge about the “true” ozone profile, which is needed as the MLP training target, is not derived from another GOME retrieval algorithm, but rather from collocated, highly accurate ozone measurements taken from different moments in GOME's lifetime and geographical coverage. These measurements stem from ozonesondes provided by the World Ozone and Ultraviolet Radiation Data Center (WOUDC) [Hare et al., 2004] and the Southern Hemisphere Additional Ozonesondes (SHADOZ) campaign [Thompson et al., 2003], as well as from the Polar Ozone and Aerosol Measurement III (POAM-III) [Lumpe et al., 2002], Stratospheric Aerosol and Gas Experiment II (SAGE-II) [Wang et al., 2002] and Halogen Occultation Experiment (HALOE) [Russell et al., 1993] occultation sounders.
 The data used for the training of this NN are similar to those that are the basis for the climatologies used as a priori information in the OE retrieval algorithms (described in section 2.1). For the ZSW algorithm, unlike the OE algorithms, there is no need to average them into, e.g., monthly means, thereby destroying information. The ill-posed problem facing classical retrieval schemes is circumvented through the use of the nonspectral input data. In particular, in areas where there is little information from the satellite spectra (compare section 3.5), the MLP automatically estimates the ozone profile on the basis of its nonspectral input parameters. In other words, it behaves like a dynamical, continuous, temperature-dependent climatology, rather than a fixed a priori data set.
2.4. Data Assimilation Approach (One Algorithm)
2.4.1. General Description
 GOME is primarily used to retrieve total ozone column densities from a spectral window around 330 nm using the Differential Optical Absorption Spectroscopy (DOAS) technique. In order to derive ozone profiles and a daily global three-dimensional (3-D) ozone analysis, the column observations are assimilated into a 3-D chemical transport model (CTM). While the CTM is driven by meteorological wind and temperature fields, the GOME observations are sequentially assimilated into the model using an optimal interpolation scheme [e.g., Khattatov et al., 2000]. The vertically integrated total column contents of the model are considered as the first-guess values. The analyzed column values are then vertically distributed weighted by the corresponding (first-guess) model profile (i.e., in ozone mixing ratios). The assimilation scheme accounts for time of observation, for spatial weighting between observation and grid, and for model and observation errors. By applying this method a global synoptic 3-D ozone analysis is available every 6 hours. Unlike the other approaches, this approach does not use the profile information in the GOME spectra, which makes it an interesting addition, as it represents the complete a priori knowledge of the ozone vertical distribution considering all relevant chemical and physical processes, and the meteorological analyses.
2.4.2. DLR Algorithm
 For the assimilation approach the German Aerospace Center (DLR) uses the 3-D global CTM called ROSE/DLR. It is based on the ROSE model described in detail by Rose and Brasseur . An updated model version (2.7) was applied to generate the data presented in this paper [Thomas et al., 2003]. This model covers all relevant gas-phase stratospheric chemical processes including oxygen, hydrogen, carbon, nitrogen, chlorine, and bromine species. Heterogeneous processes on polar-stratospheric clouds and on sulfate aerosols are also included in the model. For the assimilation of GOME total ozone column observations, the model in the DLR algorithm is run with a 5.6° × 5° longitude-latitude spatial discretization, and consists of 37 equally spaced levels covering the altitude range 8–56 km. The basic time step of the CTM is 1 hour, and therefore all GOME observations within this interval are binned. Assimilation is performed using the wind and temperature fields derived from 24-hour analyses of the UKMO following Swinbank and O'Neill .
 The optimal interpolation applied for the sequential data assimilation considers the time of observation, the spatial weighting between observation and grid, the model errors, and the observation errors. At each assimilation time step, the model's volume mixing ratios are integrated to total column values, which are then interpolated to the observation space, that is the geolocation of the GOME total column observations. In a next step the observational increments (i.e., departures from the model) are determined. The linear weight matrix operator (or gain operator) transforms the resulting innovations back to the model space [Daily, 1991], which takes into account the spatial weighting and error information of both the observations and the model. In the final step, the analyzed total columns are vertically redistributed weighted by the first-guess model profiles. For this study, the model's first guess and GOME observation errors are set to 18% and 4%, respectively. Error covariances are parameterized by hyperbolic functions depending on the horizontal distance between the model grid point and the observation [Riishøjgaard, 1998]. A correction for the model bias is applied offline, which is based on zonal mean seasonal comparison results with SAGE-II data from 1996 [Wang et al., 2002]. Contrary to the other approaches evaluated in this paper, this method delivers global synoptic 3-D ozone analyses every 6 hours. For the results used in this paper, daily mean values are provided.
2.5. Summary of the Different Algorithms
 In this section we have presented four different approaches to the retrieval of ozone profiles from GOME spectra. Five of the algorithms exploit the OE approach, each of which makes distinct assumptions which have important implications for the final retrieved profile. There is no OE method with exactly the same a priori ozone profile and related error covariance matrix. The IUP and NOAA methods use relatively large a priori error covariances everywhere, and RAL only in the troposphere and upper stratosphere, while KNMI and SAO use smaller error values.
 The OE- and PTR-based algorithms, physical-based approaches, strongly depend on the accuracy of the spectral calibration, and we have seen different ways of correcting the inadequate accuracy of the spectra delivered by GDP, including both calibration and polarization corrections. Obviously, these corrections significantly impact the ozone profile retrieval, and various analyses have been performed to quantify this effect for GOME and GOME-type instruments [see, e.g., van der A et al., 2002; Bhartia, 2002]. Furthermore, the treatment of clouds, the ozone cross sections used, the inclusion of other trace gases in the fitting process, and the exploited wavelength range are tackled in different ways. In the following sections we will investigate the implications of some of these assumptions by evaluating the retrieved data products of all these algorithms.
 We have described two NN-based methods; the UTV algorithm trained with (RAL retrieved) ozone profile data of the nadir-viewing instrument GOME, and the ZSW algorithm trained with sonde data and relatively high-resolution satellite data (from limb-viewing occultation instruments). The ninth algorithm is based on assimilation of GOME ozone column data into a chemical transport model driven by meteorological analyses (wind and temperature fields).