The direct fitting approach for total ozone column retrievals: a sensitivity study on GOME-2/MetOp-A measurements

We present a sensitivity study of the direct fitting approach to retrieve total ozone columns from the clear sky Global Ozone Monitoring Experiment 2/MetOp-A (GOME2/MetOp-A) measurements between 325 and 335 nm in the period 2007–2010. The direct fitting of the measurement is based on adjusting the scaling of a reference ozone profile and requires accurate simulation of GOME-2 radiances. In this context, we study the effect of three aspects that introduce forward model errors if not addressed appropriately: (1) the use of a clear sky model atmosphere in the radiative transfer demanding cloud filtering, (2) different approximations of Earth’s sphericity to address the influence of the solar zenith angle, and (3) the need of polarization in radiative transfer modeling. We conclude that cloud filtering using the operational GOME-2 FRESCO (Fast Retrieval Scheme for Clouds from the Oxygen A band) cloud product, which is part of level 1B data, and the use of pseudo-spherical scalar radiative transfer is fully sufficient for the purpose of this retrieval. A validation with ground-based measurements at 36 stations confirms this showing a global mean bias of −0.1 % with a standard deviation (SD) of 2.7 %. The regularization effect inherent to the profile scaling approach is thoroughly characterized by the total column averaging kernel for each individual retrieval. It characterizes the effect of the particular choice of the ozone profile to be scaled by the inversion and is part of the retrieval product. Two different interpretations of the data product are possible: first, regarding the retrieval product as an estimate of the true column, a direct comparison of the retrieved column with total ozone columns from ground-based measurements can be done. This requires accurate a priori knowledge of the reference ozone profile and the column averaging kernel is not needed. Alternatively, the retrieval product can be interpreted as an effective column defined by the total column averaging kernel. This interpretation relies much less on the a priori knowledge of the reference ozone profile; however, for its validation, measurements of the vertical ozone distribution are needed. The different manners of data interpretation are demonstrated for simulated and real measurements using on-ground ozone column and ozonesonde measurements for validation.


Introduction
Ozone is an important constituent of Earth's atmosphere, and monitoring its atmospheric abundance is essential to improve our understanding on tropospheric chemistry, air quality and climate change (e.g., Guicherit and Roemer, 2000;WHO, 2003;WMO, 2014;Fuhrer, 2009;Fleming et al., 2011).For this purpose, satellite measurements in the ultraviolet (UV) part of the solar spectrum between 310 and 340 nm form a valuable tool in measuring the vertically integrated amount of ozone with global coverage.The Global Ozone Monitoring Experiment 2 (GOME-2) (Munro et al., 2015) aboard the three European sun-synchronous, polar-orbiting MetOp satellites, with two currently in operation and the third one due for launch in 2017, measures earth radiance and solar irradiance spectra in the UV, visible, and nearinfrared spectral range from 240 to 790 nm with a spectral resolution of 0.24-0.53nm and a spectral sampling of 0.11-0.22nm.It continues a long history starting with the Solar Backscatter Ultraviolet instruments (SBUV and SBUV/2) (Bhartia et al., 1996) and the Total Ozone Mapping Spectrometer (TOMS) (Bhartia and Wellemeyer, 2004) on Nim-Published by Copernicus Publications on behalf of the European Geosciences Union.
To retrieve total ozone columns from these instruments, different algorithms have been developed.Most of the algorithms employ the Differential Optical Absorption Spectroscopy (DOAS) technique (e.g., Coldewey-Egbers et al., 2005;Weber et al., 2005;Eskes et al., 2005;Van Roozendael et al., 2006;Veefkind et al., 2006;Loyola et al., 2011), which is beneficial with respect to its computational cost.For GOME-2, the operational O3MSAF/EUMETSAT (European Organisation for the Exploitation of Meteorological Satellites) ozone column product is based on the DOAS method implemented in the GOME Data Processor version GDP v4.7 (Hao et al., 2014) and is extensively validated with ground-based measurements.Antón et al. (2009) performed a validation for total ozone columns retrieved with GDP v4.2 over the Iberian Peninsula using Brewer spectrometer measurements and found a bias of −3.05 %, while Loyola et al. (2011) report a global mean bias and standard deviation (SD) of −0.28 ± 0.7 % with respect to Dobson spectrometer measurements and −1.22 ± 0.67 % with respect to Brewer spectrometer measurements for total ozone columns retrieved with GDP v4.4.A comparison with the total ozone columns from GOME (processor version GDP v4.1), SCIA-MACHY (processor version SGP v5.0), and OMI (processor versions TOMSv8.5 and DOAO3v1.0.1 and level 1b data version collection 3) showed biases of −0.8, −0.37, −1.28 %, respectively, when compared with GOME-2 (processor version GDP v4.4 and level 1b data version 4), ensuring a consistent data set (Koukouli et al., 2012).Furthermore, degradation of satellite instruments in the UV is often observed (e.g., GOME, Snel , 2000;Tanzi et al., 2000;van der A et al., 2002, andSCIAMACHY, Noël et al., 2007;Bramstedt et al., 2009).Several methods have been published to correct measured reflectances with modeled reflectances (e.g., van der A et al., 2002;Krijger et al., 2005;van Soest et al., 2005) or by comparing measured reflectances to those at the beginning of the mission after removing both solar zenith angle and seasonal dependencies (Liu et al., 2007;Tilstra et al., 2012a).Cai et al. (2012) provide an extensive analysis of both the spectral and the cross-track degradation of GOME-2 measurements with time compared with model simulations and derived an empirical radiometric correction.
As an alternative to the DOAS retrieval approach, Lerot et al. (2010) and Van Roozendael et al. (2012) proposed the direct fitting approach for the retrieval of total ozone columns.In this approach GOME UV radiance measurements are fitted using a non-linear least squares fitting algorithm to adjust a scaling of a reference ozone profile.During the iteration the vertical profile is adjusted on the basis of the total-ozone-column-classified TOMS climatol-ogy (e.g., Bhartia and Wellemeyer, 2002;McPeters et al., 2007).An important aspect of the direct fitting approach is the simulation of the GOME measurement during the iteration using a physical model based on atmospheric radiative transport.Lerot et al. (2014) adapted the tropospheric part of the climatology from the climatology of Ziemke et al. (2011) and implemented the calculation of total column averaging kernels.The direct fitting of satellite radiance measurements to determine atmospheric trace gas columns is also successfully applied in the retrieval CO columns from SCIA-MACHY measurements (e.g., Gloudemans et al., 2008), CH 4 and CO 2 columns from the Greenhouse Gases Observing Satellite (GOSAT) (e.g., Butz et al., 2011) or in CO column retrievals from simulated Sentinel-5 and Sentinel-5P measurements (Vidot et al., 2012).
In this study, we investigate aspects of the direct fitting approach by using an implementation similar to the algorithm of Lerot et al. (2014).In particular we focus on two aspects: first, the relevance of forward model errors of the measurement simulations and, second, the role of the regularization due to the scaling of the reference profile.For this purpose, we present an algorithm which adapts the total column amount of the reference ozone profile, whose relative shape is kept fixed during the iteration.This modification of the approach of Lerot et al. ( 2014) is chosen to better analyze the effect of the regularization.As part of a least squares profile scaling approach, we simulate clear sky earth radiances using solar spectra, which are retrieved beforehand from GOME-2 solar irradiance measurements.The algorithm performance is evaluated by means of a validation with ground-based Brewer, Dobson and Système d'Analyse par Observation Zénithale (SAOZ) spectrometer data and collocated ozonesonde measurements considering a correction for the altitude of the validation site and the degradation of the satellite instrument.The effects of satellite instrument degradation on the earth radiance measurements are partly mitigated by a radiometric correction that we derived for the first 4 years of the mission from global cloud-free measurements referenced to 2007, which is the first year of the mission.Moreover, we address forward model errors such as an error introduced by residual cloud contamination in a clear sky retrieval due to poor cloud filtering, the representation of Earth's sphericity and the appropriate choice of the radiative transfer solver, considering vector and scalar radiative transfer solvers.Finally, we highlight the aspect of regularization using the total column averaging kernel to interpret the error introduced by the choice of the reference ozone profile.
The paper is structured as follows: in Sect. 2 we briefly describe the forward model and the inversion scheme.Section 3 describes the radiometric correction of the GOME-2 measurements to mitigate the instrument degradation.In Sect. 4 we present the performance analysis followed by a forward model error study.Finally, we address the aspect of regularization in Sect. 5 and conclude the paper in Sect.6.

Forward model
To retrieve total ozone columns from GOME-2 measurements, a forward model F earth is needed in order to simulate GOME-2 earth radiance measurements y earth as (1) F earth is a function of the state vector x to be retrieved and the model parameter vector b, which contains additional parameters that influence the spectrum but are not altered by the inversion.The error vector e earth combines both errors in the measurement and forward model errors.
The forward model F earth requires a solar spectrum S 0 sampled on an internal, fine spectral grid.To infer S 0 from the daily GOME-2 solar irradiance measurements y sun , we set up a forward model equation analogous to Eq. ( 1) but for the solar measurement, where we assume that the solar measurement y sun can be simulated by spectral convolution of the solar spectrum S 0 with the instrument spectral response function.This yields the forward model equation for the solar spectrum (2) The matrix K ISRF represents the convolution of the solar spectrum with the instrument spectral response function and e sun is the corresponding error vector.In Eq. ( 2), the length of the observation vector y sun is smaller than the length of S 0 , so its inversion has no unique solution.Van Deelen et al. (2007) showed that the least squares minimum length solution Ŝ0 , which minimizes the length of the solution vector Ŝ0 as a side constraint, is of sufficient accuracy to simulate earth radiance measurements of the GOME mission.Following this approach using GOME-2 solar measurements as disseminated by EUMETSAT, we calculate the earth radiance measurements by where we explicitly show the dependence of F earth on the solar spectrum and omit any other dependence.Furthermore, r is the spectral reflectance of Earth's atmosphere.Equation (3) assumes the same instrument spectral response function for solar irradiance and earth radiance measurements, so the noise contribution on the inferred solar spectrum Ŝ0 is attenuated by the convolution in Eq. ( 3).In the present study we use a Gaussian instrument spectral response function with a full width at half maximum of 0.3 nm.The transfer of light through Earth's atmosphere is described by the reflectance r as part of the convolution It includes the description of inelastic Raman scattering and elastic Rayleigh scattering of solar light, where the integral kernel r(λ, λ) represents the reflection of sunlight at the incoming wavelength λ to the outgoing wavelength λ.Numerical calculations of r are very time-consuming (e.g., Landgraf et al., 2004;van Deelen et al., 2005) and thus require an approximation to keep the numerical effort of the algorithm reasonable.Based on the concept of pre-calculated ring spectra (e.g., Hoogen et al., 1999;Hasekamp and Landgraf, 2001;Lerot et al., 2014), we approximate Eq. ( 4) by where r Ray is the monochromatic earth reflectance due to atmospheric Rayleigh scattering.It is calculated online employing the LINTRAN radiative transfer model (Landgraf et al., 2001;Walter et al., 2004;Hasekamp et al., 2005;Schepers et al., 2014).For all simulations, we use ozone cross sections by Brion et al. (1993), which are calculated for the given temperature profile, as well as scattering cross sections and phase matrices for Rayleigh scattering described by Bucholtz (1995).LINTRAN comprises a scalar and vector radiative transfer solver in plane parallel geometry and its pseudo-spherical extension.In this study, we employ the scalar solver with the pseudo-spherical approximation if not mentioned differently.Additionally, r LUT Ray and r LUT Ram are precalculated reflectances stored in a lookup table, which includes Rayleigh scattering and inelastic Raman scattering, respectively.The lookup table is calculated with the model by Landgraf et al. (2004) for the US standard atmosphere (NOAA, 1976) assuming a nadir viewing geometry, solar zenith angles between 10 and 80 • in 10 • steps, a Lambertian surface albedo A s = 0.1, and total ozone columns between 280 and 400 DU in steps of 20 DU.Here, the solar zenith angle dependence of the lookup table is chosen because atmospheric Ring spectra depend significantly on this parameter (see e.g., Joiner et al., 1995).The lookup table also includes the dependence on the total amount of ozone to avoid interference of the filling-in of ozone absorption features in the Huggins band due to atmospheric Raman scattering with the retrieved total column of ozone.Factor a in Eq. ( 5) is a free model parameter to adjust the effect of Raman scattering in the retrieval.
The use of the reflectance lookup tables r LUT Ram and r LUT Ray in Eq. ( 5) instead of pre-calculated Ring spectra bears the advantage that the simulation is based on one solar spectrum, which eases the spectral calibration of the forward model.Assuming the molecular spectroscopy of ozone as spectral reference, the forward model can be spectrally adjusted by shifting the solar spectrum in Eq. ( 5), Ŝ0 (λ) → Ŝ0 (λ+ λ S ), to account for the Doppler shift and furthermore to serve as a means of spectral calibration of the solar spectrum.A corresponding spectral adjustment of the instrument spectral response function s(λ, λ) → s(λ + λ ISRF , λ) accounts for a spectral shift between absorption features of the earth radiance simulation and the GOME-2 measurement.Both spectral shift parameters λ S and λ ISRF are independent elements of the state vector and are determined by the inversion module, which is discussed in the next section.
To simulate the sensitivity of the modeled spectrum to the parameters of the state vector, we use the linearization capability of LINTRAN, which provides the derivatives of r Ray with respect to absorption and scattering parameters of the model atmosphere.We approximate the derivatives ∂F earth /∂τ j with respect to the absorption optical thickness τ j in model layer j by and a corresponding expression ∂F earth /∂A s holds for the derivative with respect to the spectral Lambertian surface albedo A s .The derivatives with respect to the amplitude a in Eq. ( 5) can be calculated analytically in a straightforward manner.Moreover, for the Gaussian instrument spectral response function, the derivative ∂F earth /∂ λ ISRF can be derived analytically, whereas the derivatives ∂F earth /∂ λ S are determined numerically using finite differences.

Inversion module
For the inversion, we need to linearize the forward model around an initial guess of the state vector x 0 , where K = ∂F earth /∂x is the Jacobian matrix and y = y earth − F earth (x 0 , b) + Kx 0 .To retrieve the total ozone column, we follow the profile scaling approach used by Lerot et al. (2010).The state vector consists of the total ozone column c, the surface albedo A 0 at a reference wavelength λ 0 and its spectrally linear dependence δA, with A s (λ) = A 0 + δA(λ − λ 0 ), the amplitude a in Eq. ( 5), a spectral shift of the solar spectrum λ S , and a spectral shift of the instrument spectral response function λ ISRF in Eq. (3).Here, the column density c is defined by vertical profile integration, where C = (1, . .., 1) represents the corresponding geometric integration assuming an ozone profile ρ given in partial column densities per model layer.We apply Eq. ( 7) to invert Eq. ( 1) in an iterative way with respect to the state vector x using Gauss-Newton iteration, for which the minimization problem is solved in each iteration step.Here • 2 represents the L 2 norm and S e is the measurement error covariance.For this purpose, the Jacobian with respect to a scaling of a reference profile is calculated from corresponding derivatives with respect to an altitude-resolved ozone profile ρ, Here, K prof ij = ∂F earth,i /∂ρ j = σ i ∂F earth,i /∂τ j describes the profile Jacobian at a wavelength i, where ρ j is the ozone subcolumn in model layer j and σ i the corresponding ozone absorption cross section.Furthermore, ρ ref is the reference profile used for the scaling approach.From Eq. ( 10) it is clear that the profile scaling approach relies on an altitude-resolved profile Jacobian.A direct analytical calculation of the derivative ∂F j /∂c is not possible due to scattering and the temperature dependence of the ozone absorption.K col together with the derivatives of the measurement with respect to the other elements of the state vector defines the column of the least squares Jacobian K lsq and the solution of Eq. ( 9) is given by with the gain matrix The least squares scaling approach can be interpreted as a regularized retrieval of the vertical ozone distribution (Borsdorff et al., 2014).Hence, the retrieved column ĉ represents an estimate of an altitude-weighted integration of the true ozone profile, namely where A col is the total column averaging kernel, ρ true is the true ozone profile and e col is the error on the retrieved column due to the measurement error e y .This means that generally the retrieved column ĉ should be interpreted as an estimate of the effective column The part of the true column that cannot be inferred from the measurement, namely belongs to the effective null space of the inversion and is also known as smoothing error of the retrieval (Rodgers, 2000).Borsdorff et al. (2014) discussed the meaning of e n in terms of the profile scaling approach.Interpreting the effective column c eff as an estimate of the true column, e n represents the error made by the choice of the reference profile ρ ref to be scaled in the inversion.Obviously, when the reference profile represents the correct relative vertical trace gas distribution, e n vanishes.Therefore, e n does not reflect an erroneous interpretation of the measurement but originates from a wrong a priori knowledge of the vertical ozone profile.For data interpretation and validation it is desirable to disentangle both error sources.Consequently, two different conclusions with respect to the interpretation of the profile scaling approach can be drawn: (1) aiming for an estimate of the true column, it has to be stressed that accurate a priori knowledge on the relative vertical distribution of ρ ref has to be provided.In that case, the column averaging kernel is not needed for a proper data interpretation.This interpretation is adapted by Lerot et al. (2010) and further elaborated by Lerot et al. (2014), where the reference profile is updated during the iteration using the empirical correlation between the total amount of ozone and its vertical distribution (Ziemke et al., 2011).(2) Alternatively, one can focus on the information provided by the measurement and consider Eq. ( 14) as the definition of the retrieval product, where the total column averaging kernel describes a weighted altitude integration of the vertical ozone profile.Here, the effective null space error is not part of the error budget of the product and the retrieval depends much less on a priori profile information.However, the proper data use requires detailed knowledge and application of the column averaging kernel.
The comparison of both views on the data product in the context of the product validation is one aspect of this study.For validation purposes, following the first interpretation the retrieved column can be directly compared to total ozone columns inferred from ground-based spectrometer measurements, which are recorded routinely as part of a global measurement network, while for the second interpretation the vertical distribution of ozone needs to be known.For the latter, ozonesonde measurements can be used.However, due to fewer observation sites and less frequent measurements, a corresponding validation is limited in its spatiotemporal coverage.On the other hand, the advantage of this approach is the minor dependence of the data product on the a priori knowledge of the vertical ozone distribution.Important applications, like the assimilation of the total ozone column in global and regional models, preferably deal with information purely coming from the measurements and thus try to minimize the effect of ozone knowledge originating from a priori data.For such applications, the effective column together with its total column averaging kernel forms a well-suited data product.
To infer the total column averaging kernel in our algorithm, we follow the approach by Borsdorff et al. (2014).Interpreting the profile scaling approach as a particular case of a regularized profile retrieval using Tikhonov regularization of the first order (i.e., using the first derivative of the vertical profile with respect to altitude in the regularization matrix) with an "infinitely strong" regularization, Borsdorff et al. (2014) showed that the gain matrix reduces to a gain vector g col representing the fitted ozone column.Moreover, we can extract g col from the gain matrix of the least-squares fit G lsq and calculate the total column averaging kernel For the proof of Eq. ( 16), the reader is referred to Borsdorff et al. (2014).
Figure 1 shows the total column averaging kernels for retrievals from simulated measurements.Here, the mea-

A. Wassmann et al.: Sensitivity study on GOME-2 total column ozone retrievals
surements are simulated for a cloud-free scene and an ozonesonde profile on 15 January 2009 over De Bilt, the Netherlands, representing the true ozone profile.Retrievals are performed for three different reference ozone profiles: the US standard ozone profile (NOAA, 1976); the corresponding profile extracted from the climatology by Fortuin and Kelder (1998), which provides monthly averaged climatological ozone profiles in 10 • latitude bands; and the true ozone profile itself.All are depicted in the right panel of Fig. 1 as a dashed line, dotted line, and solid line, respectively.Although the US standard and the climatological reference profile of Fortuin and Kelder (1998) peak at different altitudes with different magnitudes, one can see in the left panel that the general shape of the total column averaging kernel is largely similar.The null space contribution of the sonde profile clearly differs for the given model atmosphere and is e n = 0.12 % for the US standard ozone profile and e n = −0.95% for the climatological profile, respectively.It vanishes when the ozonesonde profile itself is used as reference profile as expected.Depending on the reference profiles, we thus can expect errors on the order of 1 % for moderate solar zenith angles when we interpret the retrieved column as an estimate of the true column.For low solar elevation at high latitude stations in winter the error can even amount to 5 %.This error source can be avoided when we interpret the retrieved column as an effective column and subsequently apply the total column averaging kernel concept for validation.In Sect. 5 we demonstrate this aspect by applying both concepts to GOME-2 measurements.

Instrument degradation
To validate the presented algorithm, we apply it to spectra recorded by GOME-2/MetOp-A between January 2007 and July 2011, disseminated by EUMETSAT.Due to the choice of the retrieval window between 325 and 335 nm, we only use data from channel 2B with the advantage of the small 80 × 40 km ground pixels.Because of the optical degradation of the GOME-2 scan mirror, the reflectance measurements are subject to a spectral radiometric error.To mitigate this error, two major approaches are reported in the literature, one based on the predictability of the measurement using radiative transfer simulations and a priori knowledge about the atmospheric state and the other based on actual measurements, assuming that the averaged radiometric signal over certain regions is constant for the same period of the year, i.e., surface albedo and observation geometry.For example, Cai et al. (2012) calculate the degradation as the relative difference between measured data and modeled data in the wavelength region between 270 and 350 nm for the tropical belt (25 • S-5 • N), where ozone varies only little, for the period between February 2007 and December 2009 in 15day intervals.For GOME, Liu et al. (2007) presented a different approach comparing measured reflectances with respect to those of a reference date.Assuming a constant mean reflectance value, they considered the mean reflectance measured between 60 • N and 60 • S as a function of time for the first and 15th day of each month.Subsequently, the mean reflectance is referenced to the value determined for 1 July 1995.To remove solar zenith angle dependency and other seasonality, two third-order polynomials in time are fitted to the data.The approach assumes a constant state of the atmosphere and does not account for long-term trends in the total amount of ozone.Pawson et al. (2014) determined an average trend of 0.08 ± 0.13 % per year for the period 2000-2013.This introduces a small uncertainty in the derived radiometric degradation.There are also several advantages of using observations only over a comparison with simulated measurements.First, no collocations of the radiative transfer input parameters with the measurements are needed and, second, more importantly, uncertainties in ozone profiles, temperature profiles, and cloud data which lead to forward model errors are avoided.In that way only the effect of the instrument degradation and of atmospheric variations remain.However, averaging daily data over a large enough region reduces the impact of the latter.
In this paper, we follow a similar approach to Liu et al. (2007), monitoring degradation at three wavelengths in our total ozone fitting window, namely 325, 330, and 335 nm.For each of the wavelengths, we consider GOME-2 reflectances for the forward viewing geometry of the descending node between 60 • N and 60 • S with minor cloud contamination of cloud fraction f cld ≤ 10 %, which is calculated by the FRESCO cloud algorithm (Wang et al., 2008) and disseminated as part of the GOME-2 level 1B product.We arrange the data in 5 • latitude bins, 2 • solar zenith angle bins, and 24 ground pixel bins representing the cross-track scan.To define the degradation δI deg for the period 2008-2011, the reflectance is referenced to the corresponding reflectance of the year 2007, which is also the first year of the mission, on the same day of the year for the same solar zenith angle bin, latitude bin and ground pixel bin.We observe a clear scan angle dependence, shown for 330 nm in Fig. 2. Here, ground pixel 1 represents the easternmost pixel and ground pixel 24 represents the westernmost pixel.One can clearly identify different rates at which the across-track degradation takes place as indicated by the color gradients.The westward pixels are subject to the most severe degradation with about 9.5 % at the end of the period under investigation, while the eastward pixels are least affected (3-4 %), which is comparable to the findings of Tilstra et al. (2012b).Cai et al. (2012) report a cross-track-dependent degradation ranging from 0.6 % to less than 2.2 % from east to west pixels at 325 nm after 3 years.Based on these findings, we correct the relative radiometric degradation of GOME-2 radiances with respect to solar measurements assuming a multiplicative error contribution (R. Snel, personal communication, 2014, SRON, the Netherlands).Since the degradation showed only little spectral dependency across our fitting window, we consider it spectrally constant.From the data of Fig. 2, we derived a corresponding degradation correction for the period 2008-2011 per scan mirror position by linear regression.It is assumed that the GOME-2 data are not affected by the degradation in 2007 and hence are not corrected.

Performance analysis
In this section, we carry out a performance analysis of our direct fitting approach.Different aspects of the forward model that impact the quality of the retrieval product such as residual cloud contamination in clear sky radiative transfer modeling, the representation of Earth's sphericity and thus the dependency on solar zenith angles, and the need of polarization in radiative transfer modeling are discussed.For this evaluation, we directly compare the retrieved column with ground-based measurements using reference ozone profiles extracted from the climatology by Fortuin and Kelder (1998).Furthermore, we use temperature and pressure profiles from the European Centre for Medium-Range Weather Forecasts (ECMWF) ERA-Interim data (Berrisford et al., 2009) as auxiliary input and define a filter to reject retrievals in cloudy atmospheres.This cloud filter is based on FRESCO cloud parameters (Wang et al., 2008), which are part of the disseminated GOME-2 level 1B product, and is discussed in detail in Sect.4.1.For a proper comparison, ground-based data have to be recorded on the same day and be spatially co-aligned.Additionally, we corrected the total column from ground-based measurements for the difference in surface elevation between the measurement site and the mean GOME-2 pixel elevation, which is derived from Shuttle Radar Topography Mission (SRTM) high-resolution digital topographic database (Farr et al., 2007) and the near-surface ozone mixing ratio approximated by the ozone reference profile.
In total, we consider ozone measurements at 36 stations and extracted the corresponding data from the World Ozone and Ultraviolet Radiation Data Centre (WOUDC; see www.woudc.org)and the Southern Hemisphere ADditional OZonesondes (SHADOZ; Thompson et al., 2007) networks.The stations are listed in Table 1 together with their location and the type of the ground-based instrumentation.For some stations, more than one ground-based instrument was operational for the examined period, which allows us to intercompare different instrumentations.Furthermore, Table 1 indicates a good coverage of validation sites in the Northern Hemisphere with an even higher density of stations in Europe.
In this section, we compare ground-based ozone column measurements with collocated GOME-2 ozone columns directly.To reduce the effect of regularization in this comparison, we use the ozone climatology by Fortuin and Kelder (1998) to extract appropriate reference profiles.The algorithm validation depends also on the accuracy of the groundbased total column measurements of ozone.Fioletov et al. (2008) indicated less accuracy of zenith sky measurements and so we exclude these measurements in our study.Moreover, Basher (1982), Komhyr et al. (1989), Basher (1994), Kerr et al. (1997), andFioletov et al. (2005) reported that a precision of 1 % for well-calibrated Brewer and Dobson instruments can be reached.However, systematic differences of about ±0.6 % between both are introduced through different temperature dependencies of the absorption cross sections at the different wavelengths used by the instruments (Staehelin et al., 2003).Furthermore, Bernhard et al. (2005) stated that a fixed stratospheric temperature and the parameterization of the ozone layer in total ozone retrievals from Table 1.Number of measurements N, biases b and error SD (σ ) of the validation for each station.Data set 1 comprises the results of the direct column comparison described in Sect. 4 using profiles from the climatology by Fortuin and Kelder (1998) as reference profiles and the filter criteria δt < 30 DU, δr < 300 km, χ 2 ≤ 2, and η cld < 0.1 are applied.Data set 2 comprises the results of the effective column comparison using ozonesonde profiles as reference and the more stringent filter criterion δt < 15 DU, while the other remain as for data set 1.

Station name
Lat. carried out a validation with Dobson and Brewer instruments and found a bias of about 2 % between the two types of measurements.Thus, differences between GOME-2 retrieval and ground-based measurements have to be considered in the view of this overall uncertainty of our validation measurements.

Data filtering
For the performance analysis and the subsequent validation we define and apply four quality criteria for data selection.First, we define a cloud filter to select GOME-2 measurements with minor cloud contamination based on an effective cloudiness parameter: where f cld is the factional cloud coverage of the observed scene, z cld the cloud top height and z ref a reference height.
The effective cloudiness parameter yields largest values for high clouds and large cloud fraction and thus describes a shielding of the subjacent atmosphere.For the numerical implementation of the cloud screening, we employ the GOME-2 FRESCO cloud product (Wang et al., 2008), which is part of the level 1B product, and assume a reference height z ref = 10 km.Second, assuming the spatiotemporal variation of ozone to be ergodic, we use its temporal variation to filter spatial heterogeneity around the validation sites.Therefore, we consider 3 days of consecutive ground-based measurements, with the second day being spatiotemporally coregistered with a GOME-2 measurement.The difference of that collocated ground-based measurement with the groundbased measurements of the preceding day and succeeding day (δt) has to be less than a threshold value.Although the ergodic assumption on the ozone variation may not hold in general and thus may lead to too strict data filtering, it is used to identify spatial heterogeneity in ozone, which is laterally transported over the observed scene.Third, measurements are assumed to be spatially co-aligned when the distance between the site of the ground-based measurements and the center of the GOME-2 pixel δr does not exceed a threshold.Finally, only GOME-2 products with χ 2 ≤ χ 2 max of the spectral fitting are considered, where Here, r meas and r mod are the measured and simulated reflectance, respectively, and N is the number of spectral measurements.For the evaluation of our data product, we choose the following quality filtering: δr < 300 km, (20) For this data filtering, we find 6861 collocations between all ground-based measurements and GOME-2 observations.

Topography correction and instrument degradation correction
Before validating our data product with ground-based measurements, we introduce two important corrections.First, we correct the ground-based measurements of the total ozone column for topographic differences between the validation site and the satellite ground pixel, despite then excluding those stations, which is commonly done (e.g., Fioletov et al., 2008).Figure 3 demonstrates the relevance of this correction for the elevated sites Izaña (∼ 2300 m a.s.l.) and Mauna Loa (∼ 3400 m a.s.l.).Here, we obtain mean biases of 0.2 and 1.1 % after correction, compared to −1.8 and −3.5 %, respectively, without elevation correction.For other stations, the elevation correction is of minor importance due to smaller differences in elevation.
Second, we consider a radiometric correction for the GOME-2 instrument degradation of the UV radiance measurements, which is known to be an important issue for data interpretation (e.g., Snel , 2000;Tanzi et al., 2000;van der A et al., 2002 andSCIAMACHY Noël et al., 2007;Bramstedt et al., 2009).We investigate the influence of the scanangle-dependent degradation with time on the retrieved total ozone columns and omit the wavelength-dependent degradation since it is small across the 325-335 nm fitting window.For that purpose, we perform a validation of retrieved total ozone columns calculated from GOME-2 measurements with and without the degradation correction, which is described in Sect. 3 for a subset of collocated ground stations.The subset comprises the stations Ankara, Churchill (Brewer MKII.026),De Bilt, Edmonton (Brewer MKII.055),Hohenpeißenberg, Hong Kong Observatory, Izaña, Naha, and Paramaribo, and is chosen such that it includes measurements at different latitudes and provides good data coverage for every single station in the investigated period.
Figure 4 (top panel) shows an improvement in the validation in the last third of the time series, covering the period from September 2009 to December 2010, of ∼ 0.5 % when the degradation correction is applied (red bars).Attributed to our approach of determining the degradation, no difference between retrievals with and without degradation correction is seen in 2007, since this year serves as reference as discussed earlier.Therefore, the relative degradation of the GOME-2 reflectance measurements δI deg is zero in 2007, seen in the middle panel which shows the mean radiometric degradation averaged over the corresponding 2-month bin in the time series.The biases seen in the top panel show more variation which might be related to the choice of the validation sites, their instrumentation and the data coverage over the period under investigation, which is shown in the lower panel of the figure.However, in the context of biases of 0.6 % between Brewer and Dobson instruments (Staehelin et al., 2003) and 2 % between SAOZ and both Brewer and Dobson instruments (Van Roozendael et al., 1998), the biases that we report are close to or within the limits of the validation.
Next, we investigate the scan angle dependency of the degradation and its influence on the retrieved product.We aggregate the data set into 6-month bins of east and west pixels by dividing between eastwards (pixel index 1-12) and westwards (pixel index 13-24) scans in order to obtain meaningful statistics.Figure 5 shows that the retrieval error increases faster for the uncorrected western pixels (light blue) than for the eastern pixels (dark blue).Comparing the uncorrected re-Figure 5. Influence of the scan-angle-dependent degradation on the total ozone column retrieval error ( ret ).Data are binned into 6-month intervals as well as east and west pixel bins, which include pixel numbers 1-12 and 13-24 (Fig. 2).The lighter colors indicate the western pixels and the darker colors the eastern pixels.Furthermore, blue colors represent the data not corrected for degradation and the orange and brown colors the degradation-corrected data.The underlying data set is the same as in Fig. 4. trievals with their corrected counterpart, western pixels in orange and eastern pixels in brown, this becomes even more obvious.The spurious features seen here in the beginning of the time series are of the same origin as in Fig. 4. Because these errors already occur in the beginning of the time series, especially the difference between west and east pixels may hint at a radiometric calibration bias of the eastward pixels.From Fig. 5 we conclude that applying the degradation correction to the west pixels improves the retrieval error in the validation with ground-based instruments from ret = −1.3% to ret = −0.6 % at the end of the investigated 4-year period, while the correction of the east pixels has a smaller effect.A difference in total ozone columns of about 1.5-2 % between the west and east pixels is also reported by (e.g., Antón et al., 2009;Loyola et al., 2011;Koukouli et al., 2012;Hao et al., 2014).Hence, the application of the degradation correction improves the validation in that interval and is expected to become even more important for the ongoing mission beyond the period that we investigated.

Forward model errors
This section focuses on the relevance of forward model errors for our retrieval product.First, we consider the effect of the remaining cloud contamination after data filtering assuming a clear sky atmosphere in the forward simulation.Subsequently, we study the effect of Earth's sphericity and its approximation in the radiative transfer model and, finally, we evaluate the effect of scalar radiative transfer approximation for the retrieval of the total ozone column.

Cloudiness
Clouds significantly affect the light path through the atmosphere.Depending on the cloud optical properties, cloud morphology and surface properties, the light path can be shortened or enhanced compared to the clear sky situation (e.g., Pfeilsticker et al., 1998).Using satellite measurements, clouds are typically characterized by effective cloud parameters, which describe the light path for the considered spectral range.For GOME-2, cloud parameters are commonly retrieved from the O 2 A band at around 760 nm (e.g., FRESCO cloud parameters; Wang et al., 2008).The adaptation of these cloud parameters for a correct light path simulation in the UV is not obvious (van Diedenhoven et al., 2007).Because of this, we decided to rely on clear sky observations requiring a cloud filtering of the GOME-2 observations in the current version of our algorithm and proposed the cloud filtering of GOME-2 observations based on the cloudiness parameter η cld , defined in Eq. ( 17).To demonstrate the validity of this filtering, we consider the ozone column retrieval error as a function of the cloudiness parameter η cld and discard the cloudiness threshold set in Eq. ( 22).In this way we obtain 9600 valid data points.Next, we determine a bias from nearly cloud-free scenes (η cld < 0.1) for each station and correct the corresponding data set for it.The determined bias reflects other error sources which may depend on the particular site and are in this way disentangled from the error due to cloudiness.This correction varies between ±2 % depending on the validation site, which is also reflected in the stationto-station bias variation in the left column of Table 1.Apply- ing this correction highlights the dependence of the retrieval bias on cloudiness η cld as shown in Fig. 6.For η cld ≤ 0.1 the relative dependence on cloudiness is weak but increases significantly for η cld > 0.1, showing already a retrieval error of about −1 % for η cld = 0.15 and further increases to ret = −3.2% for η cld = 0.5.This justifies the use of the clear sky radiative transfer simulation in conjunction with cloud filtering as proposed in Eq. ( 22) in Sect.4.1.

Earth's sphericity
For satellite observations at large solar zenith angles, the treatment of Earth's sphericity as part of the radiative transfer simulation becomes an important aspect.To select a proper approach for our retrieval problem, we investigated the retrieval performance as a function of solar zenith angle for three different approximations: plane-parallel approximation, the air mass correction of Kasten and Young (1989) and the pseudo-spherical approximation (Walter et al., 2004).Here, we selected validation sites for which the GOME-2 measurements cover at least the range of 50-80 • solar zenith angles.Similarly to the disentanglement of the cloudiness error in Sect.4.3.1,we separate the error due to the influence of the solar zenith angle from other error sources.To do so, each station of the data set is corrected for its mean bias determined from solar zenith angles θ < 55 • , which varies between −2 and 3 %.In this way, we consider the relative error at larger solar zenith angles.Figure 7 shows a clear improvement when using the pseudo-spherical approximation instead of the plane-parallel approximation with and without air mass correction.For θ > 70 • , using the plane-parallel approximation underestimates the ozone column up to a mean error of 7.5 % at θ = 85 • .Errors are reduced by more than a factor 2 using the air mass correction and reduce to about −0.5 % for pseudo-spherical approximation.This is in agreement with the sphericity effect studied for simulated measurements and suggests using the pseudo-spherical approximation for our retrieval.
Nevertheless, the relative error shows some suspicious features, e.g., the positive error of 2 % for the pseudo-spherical simulations at θ = 77 • .This may be caused by the fact that the underlying data set of this study consists of Brewer, Dobson, and SAOZ measurements with different solar zenith angle dependencies.Due to the sampling of different solar zenith angle ranges of the measurements at different stations, features can be introduced.For a better comprehension of the satellite solar zenith angle dependence of the retrieval, Fig. 8 shows the retrieval error as function of the solar zenith angle for the Praha Dobson spectrometer (left) and the Praha Brewer MKIV.098 spectrometer (right).For both data sets, we identify a potential trend by linear regression.The SD of the data points with respect to the regression is used to characterize the overall quality of the regression.Although being situated at the same validation site and hence GOME-2 covers the same range of solar zenith angles, the data sets show different dependences.The Dobson instrument shows a clear positive trend with increasing solar zenith angle of 1 % per 10 • solar zenith angle, whereas for the Brewer instrument such a trend is not present in the data (0.1 %).Thus for Praha, we conclude that the error trend is probably inflicted by the ground measurements and not by the GOME-2 data, so one may suggest that the Praha Dobson spectrometer is more susceptible to solar zenith angle dependencies than Brewer instruments, which is also shown by Bernhard et al. (2005).Table 2 summarizes the slope of the regression and the SD for all stations of Table 1 with sufficient data coverage.Significant slopes are observed for the stations Churchill (B-MKIV.032),Goose Bay, Praha (Dobson), Boulder, Tateno, and Ushuaia, which is confirmed by the small variation of the SD of the data points around the linear regression within the data set.The trends in Table 2 indicate a significant error dependence on solar zenith angle for Dob-  1 for more details about the different sites).For each measurement site, the data are corrected for an overall bias for solar zenith angles θ < 55.Table 2. Linear dependence of the retrieval error ret on solar zenith angle θ , characterized by the slope m of a linear regression through the data point per 10 • of solar zenith angle (see also Fig. 8) and the SD (σ ) around the linear regression.

Station name
Lat.

The scalar radiative transfer approximation
Due to multiple scattering of solar light in the ultraviolet spectral range, correct radiance simulations have to take the polarization of scattered light into account at the cost of numerically expensive simulations.Neglecting polarization by scalar radiative transfer introduces errors in the modeled radiances, which can be as large as 10 % depending on the scattering geometry of the singly scattered light (Mishchenko et al., 1994;Stammes , 1994;Lacis et al., 1998).Consequently, one may argue that the use of the scalar radiative transfer solver in our LINTRAN forward model potentially causes retrieval biases depending on scattering geometry and subsequently on the solar zenith angles.In the left panel of Fig. 9, the spectral error in the wavelength ranges between 303 and 336 nm and is shown for different solar geometries and comprises a strong wavelength dependence for wavelengths smaller than 320 nm for almost all investigated scattering geometries.Here, the polarization of light is governed by singly scattered light, which for Rayleigh scattering has its highest degree of linear polarization for a scattering angle of scat = 90 • .Thus, polarization affects the intensity at higher scattering orders and consequently causes an error on the simulated intensity if not accounted for.For the spectral window used in this study (325-335 nm), this error comprises mainly a radiometric offset but it also includes spectral features interfering with spectral absorption features of ozone, which is shown in the right panel for the same scattering geometry.To estimate the effect of the used scalar forward model on our retrieved ozone column product, we have generated synthetic measurements for the whole range of satellite solar and viewing geometries of the validation data set for Lerwick using the LINTRAN vector radiative transfer model.The retrieval errors in the total ozone columns caused by the scalar radiative transfer approximation are shown in Fig. 10 as function of the scattering angle in single scattering geometry.In case we fit a spectrally constant effective albedo, as shown in the left panel, the error on the total ozone column is substantial with a maximum of 4 % at a scattering angle scat = 90 • , where the maximum error is expected to be.Here, the error pattern clearly follows the radiometric offset in Fig. 9.The retrieval error induced by using the scalar version of the forward model can be further reduced by fitting a linear spectrally dependent surface albedo, which is depicted in the right panel of Fig. 10.For almost all scattering angles the error diminishes and is below 0.7 % in all cases.
Although the effect of these forward model errors is small, it is interesting to see if the use of vector radiative transfer in the retrieval of total ozone columns improves the validation with ground measurements.For this purpose, we consider the validation error ret using vector and scalar radiative transfer in our forward model for a set of validation sites.To detect changes of less than 0.7 % in our retrieval product for different stations, we have to correct again for individual biases per station to disentangle the error sources.For this purpose, we  consider validation points where the difference between using a scalar or a vector radiative transfer model sca − vect is less than ±0.1 % and assume that for these cases the error is dominated by a bias which does not depend on the particular radiative transfer solver.Subsequently, the mean bias of this subset is used to correct the entire validation set for the particular station.Finally, in Fig. 11, we consider the retrieval error ret as a function of sca − vect .The figure shows a clear correlation between the differences sca − vect and the validation errors ret .For scalar radiative transfer, the differences sca − vect are mapped nearly one-to-one to corresponding errors of the validation.The use of a vector radiative transfer model thus represents an improvement of the validation data set.For sca − vect > 0.4 %, the statistics become poor due to the fact that most validation sites are situated at latitudes higher than 50 • N. Because of the sunsynchronous orbit of MetOp, this causes an asymmetric distribution of scattering angles in our data set, which might explain the larger values of ret for sca − vect > 0.4 %.Concluding on the need of vector radiative transfer to retrieve total ozone columns from the 325-335 nm UV spectral window, the induced error of less than 0.7 %, using scalar radiative transfer, has to be viewed in the context of uncertainty requirements for this data product.For example, for the future Sentinel-5 mission, an uncertainty of less then 3-5 % is required on the total ozone column product (Ingmann et al., 2012).In this context, we conclude that the use of a scalar radiative transfer solver is justified, although more accurate retrievals of the total ozone columns can be achieved using vector radiative transfer to account for the effect of polarization.
Overall, we conclude that the use of a clear sky scalar radiative transfer model using the pseudo-spherical approximation of Earth's sphericity in combination with strict cloud filtering (η cld < 0.1) is fully sufficient.For this setup, Table 1 summarizes the validation using the direct comparison of ground-based and GOME-2 total ozone columns for all stations (data set 1), reporting the mean bias and the SD of the individual error as retrieval diagnostics for each station.On a global average, the difference between the observation modes of the ground-based spectrometer are small with a mean bias of −0.1 % and an error SD of 2.7 %. Overall, we see similar biases for Dobson, Brewer and SAOZ instruments with the largest biases for Scoresbysund (3.2 %), Macquarie Island (−2.3 %), and Marambio (3.3 %).In this analysis we do not account for the effective null space, and thus from the results of Fig. 13, we expect that the biases can be overestimated, as discussed in Sect.2.2.Lerot et al. (2014) report errors due to the shape of a priori ozone profiles in the order of 1 % for solar zenith angles < 80 • that can mount up to 4 % for solar zenith angles > 80 • .

The effect of regularization
Finally, we discuss the regularization aspect of the profile scaling approach.Interpreting the retrieved column as an estimate of the true column, we showed that the direct comparison of the retrieved ozone column with ground-based measurements is subject to null space errors due to an erroneous a priori reference profile.When the retrieval product is understood as an effective column defined by Eq. ( 14), the null space error does not need to be considered, making the validation less dependent on the a priori choice of the reference profile.In this section, we will confirm these findings by comparing our retrieved column with ozonesondes and ground-based measurements for different choices of the reference profile.
To validate the GOME-2 effective ozone column product with collocated ozonesonde and ground-based measurements, we apply the quality criteria defined in Sect.4.1.Here, we adopt the filter criteria of Eqs. ( 19)-( 22) with the exception of a more stringent filtering δt < 15 DU.This reduces scattering of the validation error at the cost of the number of validation points.
Moreover, for the effective column comparison, we employ Eq. ( 13) and, hence, we rely on measurements of the vertical distribution of ozone, represented on the vertical grid of the model atmosphere (2 km thick model layers between 0 and 60 km).For this purpose, we use ozonesonde measurements, which are extended with the climatology of Fortuin and Kelder (1998) above the sonde burst height and subsequently normalized to the total column of ozone of a collocated ground-based measurement.This accounts for both, the lack of data above the burst height and systematic errors resulting from differences in pre-flight preparation of the ozonesonde (Kerr et al., 1994;Beekmann et al., 1994Beekmann et al., , 1995;;Smit et al., 1998;Fioletov et al., 2006).Here, both ozonesonde measurements and ground-based data have to be recorded on the same day and to be spatially co-aligned to meet the quality criterion in Eq. ( 20).Additionally, we corrected the total column from ground-based measurements for the difference in surface elevation between the measurement site and the mean GOME-2 pixel elevation, and applied the degradation correction to GOME-2 measurements as well, as discussed before.In general, far fewer ozonesonde measurements than ground-based measurements are available, which limits the number of validation measurements.In total, 647 collocations have been found to evaluate the relevance of the total column averaging kernel.Figure 12 displays an example of a validation of our GOME-2 total ozone column product collocated with ozonesonde measurements and Dobson direct sun measurements at Naha, Japan.The upper panel of Fig. 12 shows the time series of the retrieved total ozone column (filled circle) and the ground-based ozone column (open circle).Here, we choose the US standard reference profile to be scaled by the retrieval.Overall, we see a good agreement between both to-tal ozone columns with the same seasonal dependence.However, a closer look reveals that the direct comparison of the ozone columns results in a negative mean bias of −2.6 % with a SD around the mean of 2.2 %.The mean bias is caused by the choice of the US standard profile as reference profile and is due to null space error, which is shown in the middle panel.The latter is estimated from the ozonesonde measurements scaled to the Dobson total ozone column and the total column averaging kernel of the individual retrievals (using Eq. 13).The null space error is −2.6 %, on average, but varies between the different soundings because of the variability of the total column averaging kernel and the ozonesonde measurements with respect to the standard atmosphere.The lower panel of Fig. 12 shows the relative retrieval error for the direct and effective column comparison.For this retrieval setting, it indicates the importance of the total column averaging kernel, where for the effective column approach the mean retrieval bias is reduced to −0.2 % with a SD around the mean bias of 1.9 %.Thus, the difference in the validation of the direct and effective column approach is due to the wrong a priori information and not due to an inadequate interpretation of the satellite measurement.For each station from Table 1 employed in the effective column comparison, we consider the mean bias as a diagnostic tool to evaluate the effect of the choice of the reference profile.Results are depicted in Fig. 13 for a set of validation stations that comprise at least a total of 15 collocations with GOME-2 retrievals in the investigated period.We demonstrate the effect of three different choices for the reference profile in the inversion: (1) the ozone profile of the US standard atmosphere, (2) climatological profiles of Fortuin and Kelder (1998), and (3) the collocated ozonesonde measurements.
Using the US standard reference profile, the mean retrieval bias varies from station to station between −0.8 and −3 % for the direct comparison.For all sites, the bias is reduced significantly for the effective column comparison with mean biases between 0.6 and −1.1 %.For climatology profiles, the performance of both approaches becomes similar with biases ranging from 0.7 to −1 %.However, the validation of GOME-2 retrievals improves significantly for Naha, Hong Kong Observatory, and Broadmeadows for the effective column approach: from −0.8, −1.3, and 1 % (direct comparison) to 0.1, 0.4, and −0.2 % (effective column comparison), respectively.This might be caused by a deficiency in clima-tology by Fortuin and Kelder (1998) for these geolocations.Finally, using the ozonesonde profile as reference profile provides identical results for direct and effective column comparison with biases from 1 to −0.9 %, because the null space contribution of the reference profile vanishes (see discussion in Sect.2.2).
Moreover, the effective column comparison results in a very similar validation for the three choices of the reference profile.For example, for Naha it varies between −0.2, 0.1, and 0.1 % for the three different reference ozone profiles.The SD of the retrieval error varies only little for the different approaches.This confirms that a proper treatment of the regularization of the profile scaling approach in the validation can reduce the dependence of the validation on the particular shape of the reference profile.For all 36 stations, we summarize the validation in Table 1 (data set 2) by giving the number of collocations, the mean error, and the error SD.Here, collocated ozonesonde measurements were used as reference profile ρ ref .

Conclusions
In this paper, we presented an extensive sensitivity study for the direct fitting approach to retrieve total ozone columns from clear sky GOME-2 measurements between 325 and 335 nm.Based on an iterative least squares method, the GOME-2 reflectance measurement is fitted by adjusting the vertical ozone profile through the scaling of a reference profile, which is input to a forward model.In addition to the retrieved ozone column and its noise estimate, we provide the column averaging kernel as a retrieval product following the method of Borsdorff et al. (2014).
To account for the degradation of the GOME-2 spectrometer, we discussed a mitigation approach based on the assumption that the cloud-free mean reflectance over a certain region and a certain observation geometry and time does not change.We determined a scan-angle-dependent degradation for the period under investigation which is solely based on GOME-2 measurements that are referenced to the corresponding day in 2007 (the first year of the mission).For the eastern pixels we found a degradation of about 3-4 %, while for the western pixels the degradation is up to 9.5 % at the end of the 4.5-year period.Based on these findings we corrected the GOME-2 measurements in the period 2008-2011 and presented the improvement of our retrieval as a function of time during the first 4 years of the mission.Furthermore, we found hints to an initial calibration bias of the instrument, an aspect which has also been reported by Loyola et al. (2011).
Inherent to the direct fitting approach is the accurate simulation of GOME-2 radiances.In this context, we investigated the effect of forward model errors of the retrieval approach.We showed that the FRESCO effective cloud product (Wang et al., 2008), which is part of the disseminated level 1B data, can be used to filter GOME-2 measurements suffi-ciently accurately for cloud contamination to justify the use of a clear sky model atmosphere for the radiative transfer simulation.Another important aspect of the forward model simulation is the representation of Earth's sphericity, and connected to that the influence of the solar zenith angle on the retrieval error.We investigated three approximations -(1) plane parallel assumption, (2) air mass correction by Kasten and Young (1989), and (3) the pseudo-spherical approximation -and found the smallest biases at large solar zenith angles (θ > 60 • ) for the pseudo-spherical approximation ranging from 2 to −1 %.Finally, we investigated the need of vector radiative transfer to simulate GOME-2 observations.Here, we showed that the use of a scalar radiative transfer model in combination with the fitting of a spectrally linear dependent Lambertian albedo can mitigate the effect of neglecting polarization and yields errors on the retrieved total ozone column < 0.7 %.With the benefit of much lower computational cost and the required uncertainties in the order of ≤ 3-5 % on the total ozone column, it is sufficient to use scalar radiative transfer together with the pseudo-spherical approximation to account for the sphericity of Earth's atmosphere.
Finally, we discussed regularization aspects of the inversion of a profile scaling approach.This allows for two different approaches interpreting the retrieved total ozone columns.First, interpreting the retrieval product as an estimate of the true column allows for a direct comparison of the retrieved column with total ozone columns from groundbased measurements but requires accurate a priori knowledge of the reference ozone profile to be scaled in the inversion.Alternatively, the retrieval product can be interpreted as an effective column defined by a vertical integration of the ozone profile using the column averaging kernel as an altitude weight.This approach accounts for the effect of the particular choice of the reference profile and, hence, both data use and validation depend less on the a priori knowledge of the relative vertical distribution of ozone.Using the pseudo-spherical approximation together with the scalar radiative transfer in our forward model to simulate clear sky measurements, we validated our retrieved column using both approaches with data from different instruments at 36 ground stations.For the direct comparison, we obtained 6861 validation measurements and found an overall bias of −0.1 % with an error SD of 2.7 %.
Furthermore, we investigated the effect of regularization of the profile scaling approach.For this purpose, we considered a set of 647 validation measurements of collocated ozonesondes, ground-based measurements and GOME-2 observations.We showed a significant dependence of the retrieval error on the choice of the reference ozone profile for the direct comparison.For example, for Naha the mean bias is −2.6, −0.8, and 0.1 % when using the ozone profile from the US standard atmosphere, the corresponding ozone profile extracted from the climatology by Fortuin and Kelder (1998), and the ozonesonde itself, respectively.This depen-dence is reduced significantly considering the effective ozone column: −0.2, 0.1, and 0.1 % for the different choices of the reference profile.This finding is also confirmed for other stations.It means that errors in the validation of the direct column are due to different a priori assumptions but not due to an erroneous interpretation of the GOME-2 measurements.To our knowledge, it is the first time that the role of the column averaging kernel is presented in this manner for real measurements of the total ozone column showing that the accuracy of the effective ozone column depends less on our a priori knowledge of the relative vertical ozone distribution.Furthermore, for the first time an error on the total ozone product assessed by validation is provided that is hardly affected by any error due to the choice of the a priori reference profile information.Future studies should show if the use of the column averaging kernel is useful for applications like the assimilation of the retrieved ozone column in a global or a regional model as is the case for other satellite trace gas column products (e.g., CO 2 and CH 4 ).
Concerning future developments of our algorithm, we envisage extending the analysis including the degradation correction for the entire mission lifetime including the application to GOME-2/METOP-B data.Moreover, we aim to increase the exploitation of the GOME-2 data set by including cloudy pixels in our data processing.For GOME measurements, van Diedenhoven et al. (2007) have shown that the combination of the UV spectral range with measurements in the O 2 A band is suited to retrieve ozone profiles for cloud contaminated measurements.We propose to investigate a similar spectral combination to retrieve the total amount of ozone simultaneously with cloud properties from GOME-2 observations.

Figure 1 .
Figure 1.Left panel: total column averaging kernel using the ozone profile of the US standard atmosphere (dashed), from the climatology by Fortuin and Kelder (1998) (dotted), and the ozonesonde profile from 15 January 2009 over De Bilt, the Netherlands (solid).The right panel shows the corresponding ozone profiles.The measurements are simulated for a cloud-free scene, the ozonesonde profile and a solar zenith angle of 70 • , a viewing zenith angle of 30 • , and a surface albedo of 0.1.

Figure 2 .
Figure2.Degradation of globally averaged reflectances with respect to reference year 2007 per small ground pixel at 330 nm in the ozone fitting window (325-335 nm).Ground pixel 1 is the easternmost pixel of a scan and pixel 24 is the westernmost pixel.

Figure 3 .
Figure 3.Time series of total ozone columns and the retrieval error.Upper panels: the retrieved ozone column (filled blue circles) and the ground-based measurements using the elevation correction (open circles) for the sites Izaña and Mauna Loa.Lower panels: retrieval error of the time series corrected and not corrected for elevation differences between satellite ground pixel and measurement site (filled blue and open triangles, respectively).

Figure 4 .
Figure 4. Top panel: time series of the total ozone column retrieval error ( ret ) with (red) and without (blue) degradation correction.Middle panel: degradation for the corresponding bin referenced to 2007.Bottom panel: data abundance for each bin.The data are acquired from collocations with Ankara, Churchill (Brewer MKII.026),De Bilt, Edmonton (Brewer MKII.055),Hohenpeißenberg, Hong Kong Observatory, Izaña, Naha, and Paramaribo.

Figure 6 .
Figure 6.Mean retrieval error as function of the cloudiness parameter η cld (Eq.17) aggregated into bins of η cld = 0.05.Upper panel: the mean retrieval error in percent.Lower panel: number of data points per bin of cloudiness.In total, the data set comprises 9600 data points.

Figure 7 .
Figure 7. Mean ozone retrieval error as a function of solar zenith angle θ for different approximations of Earth's sphericity in the radiative transfer calculation.Upper panel: the mean ozone retrieval error in percent for the plane parallel approximation (PP, light blue), air mass correction of Kasten and Young (1989) (KY, dark blue), and pseudo-spherical approximation (Walter et al., 2004) (PSPH, red).Lower panel: number of data points per 2.5 • bin of θ .The validation set comprises cloud-free measurements at Resolute, Churchill B-MKII.026,Edmonton B-MKII.055,Lindenberg, Macquarie Island, Dumont d'Urville, and Goose Bay (see Table1for more details about the different sites).For each measurement site, the data are corrected for an overall bias for solar zenith angles θ < 55.

Figure 8 .
Figure 8. Retrieval error ret as function of solar zenith angle θ for the Praha Dobson (D-Beck.070)(left) and the Praha Brewer spectrometers (B-MKIV.098)(right).The blue lines are trends determined by linear regression.

Figure 9 .
Figure 9. Relative error in the radiance simulation due to the use of scalar radiative transfer modeling for different scattering geometries.Left panel: relative radiance error δI = (I scal − I vec )/I vec for different scattering angles scat and solar zenith angles θ .Right panel: same as the left panel but zoomed in on the retrieval window.The mean error for the indicated spectral window is subtracted and reported in the figure legend.The simulations have been performed using satellite solar zenith angles and viewing angles adapted from the Lerwick validation data set for a clear sky model atmosphere and a Lambertian surface albedo of 0.1.The ozone profile has been taken from the US standard model atmosphere (NOAA, 1976).

Figure 10 .
Figure 10.Ozone column error in the radiance simulation a function of single scattering angle.The retrieval error is corrected for noise contributions.Left panel: all fit parameters of our model are used next to a spectrally constant albedo.Right panel: same as the left panel but retrieving a spectrally linear dependent albedo.The simulations have been performed using the same data set as in Fig. 9.

Figure 11 .
Figure 11.Effect of the radiative transfer solver on ret .Upper panel: total ozone column retrieval error as a function of difference between scalar and vector approach.Lower panel: number of validation points.The analysis is based on measurements at Lerwick, De Bilt, Churchill B-MKIV.032,Goose Bay, Hong Kong Obs., and Izaña.

Figure 12 .
Figure 12.Time series of GOME-2 retrievals validated with ground-based Dobson direct sun measurements at Naha, Japan.Upper panel: retrieved GOME-2 total ozone column (filled circle) and the Dobson ground-based ozone column (open circle).Middle panel: the null space contribution e n .Lower panel: the retrieval error for a direct comparison of the GOME-2 column with the Dobson column (open triangles) and for the effective column comparison accounting for the effective null space contribution e n (Eq.15) (filled triangles).

Figure 13 .
Figure 13.Mean error of the retrieved ozone column using different reference ozone profiles ρ ref : (upper panel) adapted from the US standard atmosphere, (middle panel) from the ozone climatology of Fortuin and Kelder (1998), and (lower panel) collocated ozonesonde measurements.The red bars indicate the validation concept of the total column estimate including the total column averaging kernel and purple bars denote the direct comparison concept.
No observation mode was given, so all available data that meet the collocation criteria were used.b In data set 2, collocated ozonesondes from the station Hilo are used.
Van Roozendael et al. (1998) both seasonal and solar zenith angle dependencies of the ground-based measurements of up to 4 %.For SAOZ instruments,Van Roozendael et al. (1998)