We present a quasi three-dimensional ozone data set (Candidoz Assimilated Three-dimensional Ozone, CATO) with daily resolution and covering the period 1979–2004. It was reconstructed from two-dimensional total ozone observations of the Total Ozone Mapping Spectrometer (TOMS), Global Ozone Monitoring Experiment (GOME), and Solar Backscatter UV (SBUV) satellite instruments by assimilating the measurements into an equivalent latitude (ϕE)–potential temperature (θ) framework. The statistical reconstruction method uses the fact that total ozone columns are influenced by transient north-south excursions of air parcels associated, for instance, with Rossby waves. These adiabatic motions change the thickness of individual isentropic layers and transport high or low ozone into the column depending on meridional concentration gradients. A Kalman filter was employed to calculate sequentially the ozone distributions on a (ϕE, θ) grid that best match the total ozone observations given the errors in both the measurements and the ozone field predicted by the filter. CATO is shown to agree in midlatitudes within 20% with ozonesondes and the Halogen Occultation Experiment (HALOE) capturing both seasonal and interannual variations. Significantly larger differences of about 30% are found at the high-latitude sonde station Sodankylä (67°N), while differences from HALOE are mostly within 20% at these latitudes. Differences of about 30% from both the sondes and HALOE are found in the tropical lower stratosphere. This new data set is unique in its temporal and spatial coverage and will be particularly useful for studies of the factors influencing the long-term evolution of ozone in the lower stratosphere.
 Studies of long-term trends in stratospheric ozone have shown that since the late 1970s significant reductions have been observed not only over the Antarctic but also over the Arctic and the midlatitudes (see Staehelin et al.  for a review of ozone trends). Trends at midlatitudes are probably caused by a combination of effects including dynamical factors [Hadjinicolaou et al., 2002; Koch et al., 2002], dilution with polar air masses depleted in ozone [Knudsen and Groß, 2000; Konopka et al., 2003], losses by in situ homogeneous chemistry, e.g., in the upper stratosphere [World Meteorological Organization, 2003], and heterogeneous losses of ozone on aerosols below about 25 km [Solomon et al., 1996], but the relative contributions are not well quantified yet. A large number of these trend studies were based on satellite observations of total ozone columns (TO3) from the series of Total Ozone Mapping Spectrometer (TOMS) instruments or a combination of TOMS and Solar Backscatter UV (SBUV) or Global Ozone Monitoring Experiment (GOME) measurements available since 1979 [Stolarski et al., 1991; McPeters et al., 1996a; Bodeker et al., 2001]. Studies of the evolution of the vertical distribution of stratospheric ozone, however, were mostly restricted to single station Umkehr and ozonesonde measurements [World Meteorological Organization, 2003]. Studies with global coverage are comparatively scarce due to a lack of long time series of sufficiently high quality and were mostly restricted to the middle and upper stratosphere above 20 km where reliable data from SAGE I/II are available [Wang et al., 1996; Hood et al., 1993; Randel et al., 1999; Stolarski and Randel, 1998]. Wang et al.  presented an improved algorithm for ozone profile retrievals from SAGE II below 20 km which allowed them to calculate ozone trends for the period 1985–1999 down to 10 km altitude. In another study Randel and Wu  combined TO3 data from TOMS with SAGE I/II profiles above 20 km to infer ozone concentrations between the tropopause and 20 km altitude and in this way were able to calculate ozone trends as a function of altitude and latitude throughout the stratosphere. As an alternative to SAGE, a carefully homogenized version 8 of SBUV and SBUV/2 data has become available recently with better temporal and spatial coverage than SAGE but with lower accuracy, lower long-term stability, and reduced vertical resolution and extent.
 Stratospheric ozone is highly variable especially in the lower stratosphere where most of the ozone resides, and therefore long-term trends are difficult to assess [Weatherhead et al., 2000]. Statistical significance of trend estimates depends critically on how well the most important factors influencing ozone variability such as the QBO, the 11-year solar cycle, volcanic eruptions, and various modes of dynamical variability can be accounted for. Reliable estimates of these influences can only be obtained if sufficiently long and stable observation time series in the lower stratosphere are available and it is the main goal of this study to provide such a data set.
 Short-term variability, which makes up a large fraction of the overall variability, is dominated by the action of planetary-scale waves displacing air parcels from their mean position in both the north-south and the vertical (in pressure coordinates) direction. The dependence of TO3 on dynamical effects was already noted by Dobson and Harrison , who observed a correlation of TO3 with the presence of low- and high-pressure systems. As opposed to the slow diabatic Brewer-Dobson circulation, these transient air motions are adiabatic to first order and hence both potential vorticity (PV) and potential temperature (θ) are approximately conserved [Hoskins, 1991]. Owing to this conservation property of PV and its monotonically increasing values toward the poles (in a mean sense) it is possible to replace latitude and longitude by a single coordinate, the potential vorticity–equivalent latitude (PV-EL, in the following represented by the symbol ϕE). The equivalent latitude for a given PV value is the geographical latitude enclosing the same area as the PV contour associated with that PV value. The concept of PV equivalent latitude was inspired by the work of McIntyre  and McIntyre and Palmer  and was introduced by Butchart and Remsberg . Subsequently it was applied in numerous studies to reconstruct synoptic maps of constituents from incomplete observations from aircraft [e.g., Schoeberl et al., 1989; Lait et al., 1990; Hoor et al., 2004] and to extend the meridional coverage of satellite observations [e.g., Wang et al., 1999; Prados et al., 2003]. Potential vorticity coordinates have also been used in studies of the long-term evolution of total column ozone [Randel and Wu, 1995; Bodeker et al., 2001]. However, when gridding TO3 measurements by equivalent latitude, a choice needs to be made of which potential temperature level to use noting that the equivalent latitude is representative of a single isentropic level only. While the selected level may be centered close to the bulk of the ozone layer over some region, and therefore representative of most of the transport effects on the ozone column, this is unlikely to be true over the whole globe. A more comprehensive approach was followed recently by Wohltmann et al. , who developed a dynamical proxy based on equivalent latitudes at multiple isentropic levels which can explain a large proportion of short-term variability in total ozone columns observed from ground-based Dobson instruments.
 Here we present a new data set of the vertical and pole-to-pole distribution of stratospheric ozone in a (ϕE, θ) coordinate system. The data set was statistically reconstructed from satellite TO3 observations by using meteorological information interpolated to the vertical profiles above each observation point and by adopting a Kalman filter for the sequential assimilation of the observations. This method is fundamentally different from previous reconstructions which used the relation between a tracer and ϕE measured in a limited domain to extend it to a broader region based on the large-scale PV distribution obtained from meteorological analyses. In our method the stratospheric ozone distribution is obtained as an optimal solution minimizing the least squares distance between observed TO3 columns and the corresponding columns of the reconstructed distribution integrated in (ϕE, θ) space.
 The method is also different from the assimilation of TO3 columns into a chemical transport model (CTM) [Jeuken et al., 1999; Eskes et al., 2003]. The vertical distribution of ozone in these studies is essentially determined by the chemistry and transport characteristics of the model. The representation of vertical transport, however, is hampered by a number of problems related to interpolation procedures, numerical diffusion, mass conservation algorithms [Bregman et al., 2001, 2003], and problems of the meteorological analyses driving the CTMs, especially of the recent ECMWF 40-year reanalysis (ERA-40) data set [van Noije et al., 2004]. These problems are avoided with our method which provides the vertical distribution without the need for calculations of transport. In addition, it is computationally much cheaper and therefore allows us to implement the Kalman filter equations without simplifications. Assimilations of vertically resolved satellite measurements of long-lived trace gases based on Kalman filtering have recently been presented by Ménard et al.  and Khattatov et al. . These calculations are computationally very expensive and are restricted to the temporal (and vertical) coverage of the satellite data which is more limited than in the case of TO3 observations.
 Our reconstructed data set is named CATO (Candidoz Assimilated Three-dimensional Ozone) and currently covers the period January 1979 to December 2004. It is based on an update of the NIWA assimilated total ozone database of carefully homogenized TOMS, GOME and SBUV/SBUV-2 measurements [Bodeker et al., 2001, 2005] and is intended to be used for long-term trend studies. CATO is a product of the EU project Chemical and Dynamical Influences on Decadal Ozone Change (CANDIDOZ) which aims to better understand the various processes influencing variability and trends in stratospheric ozone and to find first signs of the effectiveness of the Montreal Protocol. The data set is quasi three-dimensional for two reasons. First, the full three-dimensional distribution on a given day can be reproduced if the isentropic PV (and hence the ϕE) distribution is known. Second, as a side product the latitude-longitude distribution of tropospheric ozone columns is also obtained as a residual. This potentially attractive data set is not explored further in the present study.
 The data sets used for the reconstruction and validation are presented in section 2. The physical framework and the assimilation method is described in section 3. The main characteristics of the data set and a comparison with ozonesondes and measurements from the Halogen Occultation Experiment (HALOE) will be presented in section 4. The application of a multiple linear regression model to the data set will be presented in a second study which is currently in preparation.
2. Description of Data Sets
2.1. Data Sets Used for the Reconstruction
 We use an updated version of the assimilated NIWA total ozone data base in which TO3 measurements from the series of 4 TOMS, 1 GOME, and 4 SBUV instruments are combined to a homogeneous data set [Bodeker et al., 2001]. Comparisons with the global ground based network of Dobson spectrophotometers were used to remove offsets and drifts from the individual data sets. Changes from the previous version are described in detail by Bodeker et al.  and can be summarized as follows: (1) TOMS version 7 data have been replaced by the new version 8 (available for Nimbus 7 and EP-TOMS but not for Meteor-3 and ADEOS TOMS), (2) SBUV and SBUV/2 version 8 data have been added to fill in gaps in the previous combined TOMS and GOME data set, and (3) the data set has been extended to the end of 2004.
 The NIWA total ozone data set combines the advantage of global coverage of satellite observations with the advantage of long-term stability of the ground stations. It has nearly complete temporal coverage for the period November 1978 to December 2004. The daily data are provided on a regular latitude × longitude grid at 1.0 × 1.25° resolution. On the basis of satellite orbit information (equator crossing times) we have further divided each daily sample into four subsets corresponding to the 6-hour time intervals centered around 0000, 0600, 1200, and 1800 UTC. For the 0000 UTC interval (2100–0300 UTC) data records from two different days had to be combined. An example of the TO3 observations assimilated during a single 6 hour time interval is shown in Figure 1.
 Correction functions have been applied to the Meteor-3 and ADEOS TOMS (Version 7), GOME (Version DLR 3.1), and SBUV and SBUV/2 (version 8) data in order to match the Nimbus 7 and EP-TOMS data during periods of overlap. The Nimbus 7 and EP-TOMS data sets themselves were corrected for offsets and drifts based on comparisons with the Dobson instruments. Since the new version 8 TOMS data show much better agreement with the ground based observations only small corrections had to be applied. A degradation of the EP-TOMS instrument in the last years led to an increasing drift relative to the ground observations starting in 2002 and has been accounted for in these corrections. Further information on the data sets used, the homogenization procedure, the satellite periods, and the order of preference when more than one satellite is present is given by Bodeker et al. .
 A cloud mask was calculated based on a comparison between the actual reflectivities (from http://toms.gsfc.nasa.gov/reflect/reflect.html) and climatological monthly ground reflectivities at 380 nm from Herman and Celarier  (from ftp://toms.gsfc.nasa.gov/pub/surface_reflectivity/) in order to subset pixels that are mostly cloud-free. The usage of a cloud mask has only a minor influence on the reconstructed ozone field but it improves the quality of the assimilation and the speed. Tests for the month of January 1985 have shown that by applying a cloud mask the standard deviation of the innovations (difference between measured and predicted ozone columns, see section 3 for details) can be reduced by about 10% (from 10.5 DU to 9.5 DU) reflecting the lower accuracy of TOMS TO3 values in cloudy regions where a climatological ozone profile is assumed to complete the column below cloud top [McPeters et al., 1996b]. Pixels were classified as cloudy when the reflectivity exceeded a value of 4 times the ground reflectivity. At polar latitudes a lower threshold of 1.2 was used because differences between cloud and ground reflectivities are very small and often indistinguishable when the surface is covered by ice. This method is somewhat different from that applied by Ziemke et al. , who used a fixed reflectivity of 0.2 to identify cloud-free pixels irrespective of the actual ground albedo. However, since ground reflectivities over snow- and ice-free areas are of the order of 0.05 (about 0.06 over the oceans and about 0.02–0.04 over the continents) the two approaches give very similar results over these regions. An example of a cloud mask is shown in Figure 1.
 The reconstruction method was found to generate an unrealistic ozone distribution in the upper stratosphere. The reason for this is that the chemical lifetime of ozone decreases rapidly with altitude [Duetsch, 1973; Cariolle and Déqué, 1986] such that ozone cannot be considered a passive tracer over the timescales of adiabatic motions at these altitudes. We have therefore constrained the upper stratospheric ozone concentrations using two different methods. In a first simulation ozone above 820 K (approx. 30 km altitude) was relaxed to the climatology of Fortuin and Kelder  (see section 3.4 for details). In a second simulation ozone at these levels was constrained by the assimilation of vertical profiles of the recent Version 8 SBUV data set obtained from http://www.cpc.ncep.noaa.gov/products/stratosphere/\sbuv2to/. Profiles were only included where the data quality control flags in the SBUV data files were 0, 10, 100 or 110. Here we will only report on the first simulation since differences are only small below 820 K which is the focus of the present paper. Differences become more important when studying long-term trends and variability in stratospheric ozone since the relaxation to the climatology is damping any long-term changes in the upper stratosphere.
 As meteorological input we use the 6-hourly data of the recent European Centre for Medium Range Weather Forecast (ECMWF) 40-year reanalysis (ERA-40) for the period of November 1978 to August 2002 and of the operational ECMWF analyses thereafter. ERA-40 fields were obtained from the Swiss National Supercomputing Center CSCS where they have been mirrored for better accessibility to the Swiss climate science community. Potential vorticity fields were calculated globally on a 1° × 1° grid at 60 hybrid levels and then interpolated onto 16 potential temperature (θ) levels between 320 K and 1445 K. Similarly, pressure fields were interpolated onto the θ layer interfaces (half levels) to determine the thickness of each layer required to convert between volume mixing ratios (VMR) and Dobson units (DU) per layer. PV fields were finally converted into equivalent latitude (ϕE) distributions by calculating the relationship between modified PV as defined by Lait et al.  and ϕE for each 6-hour time interval and θ level separately.
2.2. Data Sets Used for Validation
 We have compared CATO with the latest version 19 data of the Halogen Occultation Experiment (HALOE). The HALOE instrument onboard the Upper Atmosphere Research Satellite (UARS) uses solar occultation to measure an ozone profile twice per orbit, one at sunrise and one at sunset, yielding approximately 30 profiles per day [Russell et al., 1993]. The profiles are obtained at varying longitudes and at the two latitudes of the sunset and sunrise events which are changing only slowly between about 70°N and 70°S over the course of a month. HALOE has provided stable profiles at high vertical resolution (∼2 km) since its launch in September 1991. We have used only the measurements of the years 1998 to 2000 for validation because we have used a computationally expensive method to map HALOE data onto an equivalent latitude–potential temperature grid.
 Ozonesonde measurements were mostly obtained from the World Ozone and UV Data Center (WOUDC), Canada. We have selected four stations for the comparison with CATO: a southern midlatitude station (Lauder, New Zealand), a tropical station (Hilo, Hawaii), a northern midlatitude station (Payerne, Switzerland), and a northern high latitude station (Sodankylä, Finland). Only measurements of reasonably good quality were selected following the recommendations of Logan  for the range of acceptable correction factors for Brewer-Mast and ECC sondes, and using only balloon ascents extending beyond 25 km altitude. The Hilo data were first obtained from WOUDC but then replaced by an updated version available through the Climate Monitoring and Diagnostics Laboratory (CMDL) of the National Oceanic and Atmospheric Administration (NOAA) (Samuel Oltmans, personal communication). In this updated version SBUV observations are used to complete the profiles above the balloon burst level. All ozonesonde measurements are normalized to concurrent Dobson TO3 observations except for those at Sodankylä because normalization is not possible during the winter months for this high-latitude station. Correction factors for other months are mostly within the range 0.9 to 1.1 (mean value 1996 to 1999 is 1.035) such that differences would be quite small if the corrections were applied.
 Finally, we have used ozone fields provided as part of the ERA-40 data set for comparison with CATO. This comparison has been limited to the year 1998 and to profiles at the position of the ozonesonde station Payerne. Ozone is fully integrated into the ECMWF forecast model and analysis system as an additional three-dimensional variable similar to humidity. Ozone chemistry is treated in a highly parameterized way using an updated version of the Cariolle scheme [Cariolle and Déqué, 1986]. The ERA-40 reanalysis includes assimilation of TOMS TO3 columns and SBUV profiles of the layers 0.1–1 hPa, 1–2 hPa, 2–4 hPa, 4–8 hPa, 8–16 hPa, and 16 hPa to surface. See Dethof and Hólm  for further details.
3. Statistical Reconstruction Method
 The reconstruction is based on the following assumptions which were partly outlined already in section 1:
 2. Fluctuations in vertical ozone columns at a given location are determined mainly by horizontal advection of air masses in the presence of horizontal gradients in ozone through changing the volume mixing ratios in, and/or the thickness of, individual isentropic layers [Wohltmann et al., 2005].
 3. The ozone field represented in this reference frame changes only slowly with time. The chemical lifetime of ozone as well as the timescales of diabatic processes and of friction and mixing, that could change the correlation between ozone on potential vorticity [Wirth, 1993], are assumed to be much longer than the model time step of 6 hours. In a sequential assimilation we can therefore use the analysis of the previous time step directly as a first guess for the following analysis.
 4. The distribution of tropospheric ozone columns is closely related to the distribution of the sources of ozone precursors and can therefore be better described in geographical rather than in ϕE coordinates. Zonal variability is thus only permitted in the troposphere. The assumption that the ozone field is persistent from one time step to the other is also made in the troposphere although this is clearly not as well justified as in the stratosphere. However, the tropospheric ozone field is represented on a coarse grid of 10 × 6 degrees horizontal resolution at which scales the tropospheric column ozone amount is expected to be rather stable over several days.
3.1. Setup of the Reference Framework
 The quasi-three-dimensional (3-D) ozone volume mixing ratio (VMR) field χS,T to be reconstructed is composed of a stratospheric part χS represented on a (ϕE, θ) grid and a tropospheric part χT on a geographical longitude-latitude (λ, ϕ) grid. The tropospheric distribution is not resolved in the vertical but represents the entire column from the surface to the 320 K isentrope, which is the lower boundary of the stratospheric domain. The 320 K surface is close to the tropopause in midlatitudes and typically extends down to about 600 hPa in the deep tropics. It is near the lower boundary of the middle world, where isentropes intersect the tropopause [Hoskins, 1991], and it is approximately the lowest θ surface not touching the ground even in the case of high topography. Note that with the above definitions no strict separation between the troposphere and the stratosphere in the classical sense is made.
 In a typical data assimilation scheme a best estimate (or first guess) S,T of the true nonobservable field χS,T is compared with some observations to obtain a better estimate, that is a new analysis, given the additional information provided by the observations. In order to compare the distribution χS,T with a TO3 observation from TOMS or GOME sampled at a geographical location (λ, ϕ) we need to calculate the integral ozone amount along an equivalent latitude profile as illustrated in Figure 2. The ozone column is given by
The first term on the right-hand side is the tropospheric ozone column, and ΔpT is the thickness of this layer given by the difference in pressure at the surface and at θ0 = 320 K. The scaling factor a is used to convert to Dobson units: a = 0.788 DU/ppb/hPa. ϕE(λ, ϕ, θ) is the equivalent latitude profile at location (λ, ϕ).
Figure 2 also illustrates the basic principle of the reconstruction. In the example ozone-rich air is advected from the south into the column between about 800 K and 1200 K (arrow 1). Ozone-rich air is also advected at 400 K (arrow 2) but this time from the north due to the reversed meridional concentration gradient. The result is an above average TO3 column at the given location.
 To simplify the notation the discrete ozone VMR field χS,T will be represented by the state vector x in the following. The stratospheric grid is defined on 15 isentropic layers centered at θ = 326, 340, 365, 400, 445, 500, 565, 640, 725, 820, 925, 1040, 1165, 1300, and 1445 K, and on 30 equivalent latitude bands centered at −87, −81, +81, +87°N. The integral in equation (1) is then approximated by a sum over the θ layers and the discrete stratospheric field xS is interpolated linearly to the equivalent latitude of each layer. The lowest θ layer extends from 320 K to (326 + 340)/2 K and the highest layer from (1300 + 1445)/2 K to the top of the atmosphere. The tropospheric grid has a 10° longitude by 6° latitude resolution (dimension of 36 × 30). The state vector is thus composed of 15 × 30 + 36 × 30 = 1530 elements.
3.2. Sequential Assimilation Using a Kalman Filter
 Suppose that an unobservable state variable evolves according to a linear dynamic model and that we want to estimate the state variable at time i based on partial and noisy measurements up to the same time. Then the Kalman filter provides a recursive computation of the best linear unbiased state estimates (in the sense of minimizing the mean square error), together with the associated error covariance matrices. Here, the state variable is the true ozone field at 6-hour time steps, and we have the TO3 measurements (and optionally SBUV profiles above 820 K) available during the corresponding 6-hour intervals.
 The following notations are used: xi is the true (nonobservable) state, that is the O3 VMR field at time i, yi is vector of observations available in the 6-hour time interval i, i is best estimate of xi given all past and current observations y1, …, yi, and i is best prediction of yi given observations y1, …, yi−1.
 The connection between the state variables xi and the observations yi is given by
Here, each row of the linear observation operator Hi is obtained from equation (1) by approximating the integral by a sum. The observation errors ηi are assumed to have zero mean and covariance matrix Ri and to be uncorrelated for different time points. In case of a SBUV profile the corresponding row of Hi is basically composed of spatial interpolation factors which were calculated in a way that conserves the column amount in the respective layers.
 The dynamical evolution of the true ozone field is described by the following persistence equation (see assumption 3 in section 3):
Here, the random model errors εi have zero mean and covariance matrix Bi and are uncorrelated in time.
 The state update equation (or Kalman filter equation) provides the optimal analysis i of the model state at time i recursively from the previous analysis at time i − 1 given the new observations yi of time interval i:
 The vector vi = yi − i is called innovations and Ki is the Kalman gain matrix providing the weights to be attributed to the innovations and distributing them over the model domain. It is given by the following set of equations:
Pi and Ai are the first guess and analysis error covariance matrices, respectively. Equations (8) and (9) provide the recursive means for updating the matrix Pi from time i − 1 to time i. Note that the Kalman gain matrix is basically proportional to P/(R + P). The relative magnitude of the errors Pi and Ri thus determines how much weight is attributed to the first guess estimate i−1 in equation (4) relative to the observations yi. In the case of relatively small observation errors (R ≪ P) the ratio P/(R + P) approaches unity and the new analysis vector is forced to closely match the observations. In the case of relatively large observation errors (R ≫ P), however, the Kalman gain is approaching zero and the analysis is forced only weakly toward the new observations. The latter case corresponds more closely to our situation because the assumption of persistence appears to be well satisfied and the model forecast error P is therefore relatively small.
 The way the Kalman filter works is illustrated in Figure 3. It shows the standard deviation of the innovations yi − i (Figures 3a and 3b) and of the increment vectors Ki(yi − i) mapped onto the stratospheric part of the domain (Figures 3c and 3d) taken over all times i of the month October 1982. Figure 3 demonstrates how large the innovations typically are from one analysis to the next and how Ki weighs and distributes them over the different levels of the domain. In October the ozone field at high southern latitudes is recovering from the Antarctic ozone whole and therefore the innovations and increments are largest over that region. Note that in the lowermost stratosphere the absolute increments (Figure 3c) are small but due to the small concentrations here the relative changes are large (Figure 3d). The regions of largest increments correspond closely to the regions with largest uncertainties in the reconstructed ozone field (compare section 4.1). The global average standard deviation of the innovations for this month was 8.2 DU (or 2.76% of the monthly mean columns). By looking at all years of the assimilation it is found that the standard deviations averaged over the globe are highest in February/March (typically between 11 and 11.5 DU) and lowest in August/September (between 7.5 and 8 DU).
 The set of equations describing the Kalman filter (4)–(9) has been calculated recursively from November 1978 until December 2004, starting with an initial climatological ozone distribution. As shown in section 4.3, the simulation becomes virtually independent from the initial field within only a few months of integration. In periods of missing data a relaxation toward a climatology was applied as described in section 3.4.
3.3. Modeling and Optimization of Observation and Forecast Error Covariances
 Specification of the matrices Ri and Bi is a central problem in Kalman filtering [Gelb, 1974; Harvey, 1989]. We have chosen simple parametric forms such that the matrices are fully determined by a few parameters which were optimized in a separate set of model experiments described later. The observation error Ri is modelled as a diagonal matrix reflecting the assumption that the individual observation errors are independent. Note that we have subsampled the original TOMS/GOME data grid by assimilating only every third latitude band and every fourth point along each band in order to reduce computation time. The diagonal elements corresponding to observation j at time i (yi;j) are modelled as
where zai;j is the solar zenith angle of observation j and εR is a reference error to be optimized. Random errors of TOMS and GOME TO3 retrievals are of the order of 2% increasing to up to 6% at very high zenith angles [McPeters et al., 1996b]. A comparison with Dobson instruments, however, indicates an even better precision and smaller zenith angle dependence for TOMS Version 7 data [McPeters and Labow, 1996]. The latest version 8 is likely even better. Equation (10) takes into account the dependence of the retrieval error on solar zenith angle. Note, however, that Ri as defined by equation (2) not only accounts for errors in the observations but also for errors in the calculation of the projection operator Hi. These are related in a complicated manner to errors in ECMWF meteorological analyses of winds and temperature (from which PV and finally ϕE are calculated) and are possibly more important than the retrieval errors. The optimization yields a value for εR of approximately 0.02 corresponding to an error of 2% at low and 3% at high solar zenith angles.
 Our model forecast error covariance matrix Bi has positive off-diagonal entries which reflects our expectation that ozone values at short distances are changing in a similar way. This also ensures a smooth stratospheric ozone distribution in our analysis since the Kalman filter makes a compromise between the information from the observations and the information from the model forecast weighted by their respective covariances.
 The correlation length was empirically selected to provide a distribution that does not exhibit too much fine structure but at the same time preserves strong gradients across, for instance, the polar vortex edge. The coefficients corresponding to the stratospheric part of the domain are modelled as
where εS is a stratospheric reference error, ci;k is the monthly mean ozone field of the Fortuin and Kelder  climatology interpolated to the time i of the analysis and to cell k of the (ϕE, θ) grid, and v and h are weight factors. These are simple exponential functions of the indices k and l dropping off over an e-folding distance of 1.5 θ layers in the vertical and 1.5 ϕE cells (or 9°) in the horizontal, respectively. They are scaled to unity for the diagonal elements (k = l). The tropospheric part was modelled in an analog way with a reference error εT and an e-folding distance of 12° in the latitude and 20° in the longitude direction.
 The matrices Ri and Bi are thus entirely determined by the three parameters εR, εS, and εT which were optimized by systematically varying their values in a separate set of about forty simulations of the first year 1979 and by maximizing the log likelihood (LLH) function computed at the end of each simulation period. The log likelihood can be computed easily for a Kalman filter [Harvey, 1989] to
with vi being the vector of innovations and Mi their error covariances given by equation (7). In addition to the value for εR already given above the optimization yielded a value of about 0.01 for εS and about 0.08 for εT, respectively. However, the maximum of LLH is not sharply defined with respect to the ratio εS/εT and within a certain range other combinations of lower/higher εS and higher/lower εT gave nearly equivalent results. For reasons described in section 4.1 we have finally selected a somewhat different set of parameters with lower stratospheric and tropospheric model forecast errors (εR = 0.023, εS = 0.0046, εT = 0.0115).
3.4. Relaxation Toward Tropospheric Climatology
 We incorporated a slow relaxation of the tropospheric distribution toward the monthly zonal mean climatology of Fortuin and Kelder  with a time constant of 50 days to avoid the occurrence of large negative concentrations which were found if no such relaxation was applied. The reason for the tendency of the assimilation to place slightly too much ozone into the stratosphere and therefore leaving too little for the troposphere is unclear. In the simulation without assimilation of SBUV profiles we applied a relaxation not only to the tropospheric domain but also to the uppermost stratospheric layers (i.e., above 820 K) because the method fails to yield a realistic ozone distribution at these altitudes as already noted before. To enable a smooth transition and to account for the decreasing chemical lifetime of ozone the strength of the relaxation was increased with altitude by changing the relaxation time constant from about 70 days at 820 K to 2.5 days at 1445 K.
where Di is a diagonal matrix with diagonal elements equal zero where no relaxation is applied and equal 1/τ elsewhere. τ is the relaxation time in units of 6-hour time intervals and ci is the climatological field taken as the monthly means of Fortuin and Kelder  interpolated to the time of the analysis. The set of Kalman filter equations which we finally used is then obtained by modifying the remaining equations to include relaxation. Equations (4), (5), and (8) then need to be replaced by
4.1. Mean State and Variance
 An important advantage of the Kalman filter compared to other assimilation techniques is that it not only provides an estimate of the mean state but also of its uncertainty, which is an important indicator of the quality of the assimilation. As an example the ozone distribution on 15 January 1984 and its standard deviation (square root of the diagonal elements of matrix Ai) are shown in Figure 4. The stratospheric mean state and standard deviations are displayed in Figures 4a and 4b, and the respective tropospheric distributions in Figures 4c and 4d. According to Figure 4b the relative uncertainty in most of the stratosphere is between 2 and 4%. Lower uncertainties are found above 10 hPa where they are reduced by the relaxation (compare equation (16)). In the lower stratosphere the advantage of using an equivalent latitude framework is clearly visible. Despite the fact that TOMS cannot measure at high northern latitudes, the errors increase only moderately toward the pole because high (equivalent) latitude air of the polar vortex is occasionally transported to lower latitudes where it can be observed by the satellite. This is particularly true for altitudes between 150 and 40 hPa whereas both above and below the uncertainty increases more rapidly.
 The highest relative errors occur in the upper tropical troposphere where concentrations are generally low and in the lowest two model layers at high latitudes. The latter appears to be related to problems caused by variations in tropopause altitude. At midlatitudes to high latitudes the tropospheric domain (defined as the atmosphere below 320 K) may include a fraction of lower stratospheric air. At a given latitude/longitude position the stratospheric fraction varies along with changes in tropopause altitude. This results in high column mean “tropospheric” ozone mixing ratios in regions of upper level troughs as present over the Hudson Bay and over northern China on the given day (see Figure 4c). Rapid changes in tropopause altitude induce rapid changes in tropospheric ozone amounts which violates our persistency assumption 4 in section 3 and results in relatively large innovations at the given position. Since a low tropopause is correlated with advection from high latitudes in the lowermost stratosphere, the Kalman filter then assigns a part of the innovations to the model grid cells located in the lowermost stratosphere at high equivalent latitudes. This not only leads to a high uncertainty but also to an overestimation of ozone in the lowest stratospheric layer at high equivalent latitudes which appears to be compensated by too low values in the next layer above. By selecting smaller error parameters εS and εT (which reduces the weight of the observations relative to the forecast) than suggested by the log likelihood optimization this problem could be reduced to some extent, but a better description of the delicate interface between the tropospheric and stratospheric domain will be required to eliminate this problem in future simulations.
 The tropospheric distribution in the tropics shows the familiar wave-one pattern [Thompson et al., 2003] with a maximum over the Atlantic between Africa and South America and a minimum over the Pacific. Very low values (sometimes below zero) are seen over the Himalayan due to the high topography and over the Pacific at about 150°E, 20° N where ozone concentrations in the lower troposphere are expected to be very low. Note that our method assumes a zonally symmetric (in equivalent latitudes) ozone distribution above the 320 K level. As a consequence any zonal asymmetries above this level, which are known to be present in the tropical upper troposphere, must be assigned to the tropospheric distribution below 320 K. The zonal variability in the tropospheric distribution (Figure 4c) is therefore exaggerated which explains the occasional appearance of negative values over the Pacific. In a following study we will show that integration from the surface to the tropopause nevertheless yields realistic tropospheric columns.
4.2. Total Ozone in Equivalent Latitude Coordinates as Opposed to Normal Zonal Means
 Zonal mean TO3 values were calculated from the original NIWA data set in normal geographical space and compared with zonal means of CATO calculated in equivalent latitude space. These were obtained by adding the stratospheric column at a given equivalent latitude to the zonal mean of the tropospheric field at the same geographical latitude. Figure 5 shows the zonal mean CATO (grey lines) and NIWA values (black lines) as a function of (equivalent) latitude exemplarily for November 1990 (solid lines) and January 1991 (dashed lines).
 TO3 values of CATO match the values of the original data set very closely which is not surprising since it is these TO3 values that were assimilated. Deviations are very small in the tropics and become larger toward the poles. Differences between the two data sets are expected since they use different latitude metrics (equivalent latitude for CATO and geographical latitude for NIWA). An important advantage of the equivalent latitude space is that different air masses can be better separated. This is seen for instance in the November distribution where the CATO zonal means (grey solid line) display a much sharper gradient across the border of the Antarctic polar vortex than the original data set (black solid line).
 Another advantage of the equivalent latitude framework is its better meridional coverage extending beyond the geographical latitude range covered by the satellite observations. In Figure 5 the CATO data extend to the North Pole whereas the NIWA zonal means stop at about 75°N in both November and February. However, the uncertainty in the reconstructed field increases significantly toward the poles as already demonstrated in Figure 4. This effect is larger in the Southern Hemisphere winter because the Antarctic vortex is less disturbed and therefore the chance of sampling vortex air at midlatitudes is smaller than over the Arctic.
Figure 5 also illustrates that the buildup of ozone at high latitudes in the Northern Hemisphere between November and February, which is a result of the intensified Brewer-Dobson circulation during winter, is well captured by CATO.
4.3. Sensitivity to Initialization
 The sequential assimilation was initiated on 1 November 1978 with the zonal mean climatology of Fortuin and Kelder  interpolated onto the CATO grid. As shown below the influence of this initial field gradually vanishes with time as more and more observations are assimilated. In order to minimize any influence of the initial distribution we have repeated the simulation of the first year (November 1978 to October 1979) two times using the field calculated for 31 October 1979 as initial distribution for the following run. The first 2 months November and December 1978 following this 2-year cycle were then considered as additional spin-up time and only the period January 1979 to December 2004 was finally included in CATO.
 In order to test how quickly the memory of the initialization is lost we have performed two additional 2-year simulations started on 1 January 1985 with different initial distributions. The ozone fields of the full CATO simulation are used as reference. In the first simulation meridional concentration gradients were removed from the initial field on 1 January 1985 using the same layer-average volume mixing ratio at all equivalent latitudes. In the second simulation, only the vertical distribution was modified, whereas meridional gradients were preserved: Volume mixing ratios were increased by 60% at 320 K and reduced by 60% at 1445 K relative to the reference distribution with a gradual transition in between.
 The evolution of the relative difference between the reference run and the second simulation is illustrated in Figure 6. After the first month the relative differences have already decreased substantially to mostly less than 10% except for the upper tropical troposphere and the lowest stratospheric layers. Above approximately 10 hPa the relaxation supports a rapid convergence. After 6 months, small differences only remain in the upper tropical troposphere and near the tropopause at high latitudes and after 18 months also these differences have nearly disappeared as emphasized by the almost random pattern of the zero difference line. A very similar evolution of the relative differences is also seen for the first simulation in which latitudinal gradients were removed (not shown). The fact that simulations initiated with largely different distributions converge to the same solution as the reference simulation (within the level of uncertainty) demonstrates the robustness of the method and that the optimal solution is well defined. However, the simulations also reveal a relatively long memory of the initialization, especially at places such as the upper tropical troposphere where ozone is not well constrained by the method. New observations have relatively little weight compared to the first guess and therefore this new information is incorporated into the analyses only slowly. As a consequence of this the annual cycle of the ozone distribution and in particular of the shapes of the vertical profiles is most probably lagging somewhat behind the true evolution.
4.4. Analysis of Residuals
 In this section we examine the consistency of the error covariance parameterizations of section 3.3 by analyzing the residuals defined as
where Mi is given by equation (7). The residuals are thus the innovations vi ≡ yi − i normalized by their covariance matrix Mi.
 A first useful diagnostic suggested, e.g., by Ménard et al.  is the χi2 statistics defined as
 If the covariance matrix Mi, which is obtained in the course of the assimilation process, correctly describes the true variances of the innovations then we expect that χi2 is equal to the number of observations Ni used in the analysis (i.e., equal to the length of the vector yi). Actually, for a large number of observations the values of χi2 should be normally distributed with mean Ni and variance 2Ni [Ménard et al., 2000]. Figure 7 shows the temporal evolution of monthly averaged values of χi2/Ni between January 1979 and December 2004. Figure 7 confirms the consistency of the parameterizations since over the entire simulation period the values vary about the expected value of 1. Only during some months in 1995 and 1996 when GOME observations were used to fill in gaps in TOMS the values are much larger than expected. During this period the residuals are too large which is equivalent to saying that the covariances Mi are too small. Since there is no reason for assuming a different model error during this period this suggests that using the same observation errors Ri for GOME as for TOMS underestimates the true errors of the GOME DLR V3.1 data. Note that for GOME both cloud free and cloudy observations were used which may partly explain why GOME errors are on average larger. Since only a relatively short time period is affected and we expect the influence on the mean state to be small we have chosen not to repeat the simulation with an improved error estimation for GOME.
 There is also an apparent annual cycle in the ratios χi2/Ni with larger values during Northern Hemisphere winter/spring than summer/autumn. This is unlikely due to an annual cycle in the precision of the observations yi but rather due to seasonal differences in the quality of the forecasts i. Model errors thus appear to be underestimated during boreal winter/spring and overestimated during summer/autumn.
 As a second consistency test we have analyzed temporal autocorrelations of the innovations. Optimality of the Kalman filter implies that the innovations are spatially and temporally uncorrelated. In most applications, however, this can only be achieved approximately due to systematic errors in the dynamical model and due to simplifications in the design of the covariance matrices. The analysis of the correlation structure of the innovations therefore allows detecting serious errors in model and/or matrix design and provides an indication of how closely optimality is approached. Here we restrict the analysis to temporal correlations. To simplify the analysis we have performed a separate simulation over a few months in which we kept the set of observation locations constant (which required turning off the cloud mask). Then we denote by yi;m the TO3 observation at the ith 6-hour interval and position m. Because the same region of the globe is scanned only once a day, for a fixed m only every fourth observation is available. We thus consider the temporal autocorrelations ρk;m of the normalized innovations i:m (normalized with the diagonal elements of matrix M, i.e., i;m = vi;m/) for fixed m and time lags k = 4, 8, …
Figure 8 shows the statistics of ρk;m over all positions m on the globe and for lags k = 4, 8, …, 40 corresponding to 1 to 20 days. Figure 8 indicates a rapid drop to below 0.05 already on the third day suggesting that the innovations provide largely independent information to the assimilation.
 The above analysis of the innovations and residuals with respect to autocorrelations and χ2 statistics shows that our assumptions for model and observation errors are largely consistent but it also outlines options for future improvements.
4.5. Comparison With Ozonesonde Profiles
 For each balloon ascent at one of the four stations Lauder, Hilo, Payerne, and Sodankylä (see section 2.2) the potential vorticity profile was calculated and converted to a (ϕE, θ) profile. A corresponding CATO profile was then obtained by linearly interpolating the reconstructed ozone field onto the (ϕE, θ) coordinates of the sonde profile. An example of a measured and reconstructed profile is shown in Figure 9a. We have selected a case with a particularly pronounced ozone lamina in the lower stratosphere as this best illustrates the strength of our method. For comparison, Figure 9b shows an ozone profile provided by the ERA-40 reanalysis interpolated in time and space to the locations of the sounding. Figure 9 demonstrates the large influence of differential advection in the individual layers on the shape of an ozone profile as already noted by Reid and Vaughan , Orsolini et al. , and Koch et al. . The equivalent latitude profile suggests that advection from high and low latitudes is responsible for the elevated ozone values at 120–150 hPa and the reduced values at 70–100 hPa, respectively. The CATO reconstruction is able to reproduce most of the observed features. The overall shape of the profile is also reproduced in the ERA-40 data set but the amplitude of the ozone anomalies is largely underestimated.
Figure 10 shows seasonal mean profiles averaged over all ozonesondes and corresponding CATO profiles available during 1998. Seasonal mean ERA-40 profiles (average over days when a sonde was launched) were only calculated for the station Payerne and are shown in Figures 10i–10l for comparison. The main features of the observations (black solid lines) and differences between the stations are well reproduced by CATO (black dashed lines). These include the increase in the concentrations of the ozone maximum from autumn to spring (on the respective hemisphere), the concurrent descent of the altitude of the ozone maximum, the low ozone concentrations in the tropics up to about 100 hPa, and the sometimes large seasonal variations in the lower stratosphere, as seen, for instance, at Lauder.
 A comparison between different years indicates that there is also a substantial year-to-year variability in the lower stratosphere, in particular during the spring months. Figure 11 illustrates that this year-to-year variability is fairly well captured by CATO and that many of the irregular features in the profiles can be reproduced successfully. The ozone profiles of the ERA-40 data deviate significantly from the observations during winter and spring but are similar in quality to CATO during summer and autumn. This winter-spring problem is known to be the result of a too strong Brewer-Dobson circulation as well as noise in the vertical wind velocities leading to excessive vertical diffusion [van Noije et al., 2004]. Note that our reconstruction does not seem to suffer from these problems as we do not make use of vertical wind fields but only of PV distributions which are much less affected.
 There are also several notable differences between the observations and our reconstruction. The largest deviations are seen at the high latitude station Sodankylä in winter and spring. One reason for this is that the level of uncertainty generally increases toward high latitudes during winter as illustrated in Figure 4b. However, the total ozone column, which is roughly given by the area enclosed between the y axis and the vertical profile and which is determined by the TOMS observations, also seems to be lower in CATO than in the observations. In contrast to the soundings the satellite can observe an air mass originating from the polar vortex only if it is advected to lower latitudes. However, in this case the polar air is exposed to sunlight which may trigger rapid ozone destruction of the order of a few ppb per sunlit hour [Rex et al., 1997]. This may sum up to a loss of 1 mPa or more after a few days of exposure. Significant ozone losses for the winter 1997/1998 were reported for instance by Guirlet et al. . This loss would not have occurred if the same air mass (having the same equivalent latitude) would have remained at a higher geographical latitude and therefore our reconstructed ozone distribution is probably biased low over polar regions during winter in both hemispheres.
 Another notable difference is seen at the tropical station Hilo, Hawaii, where CATO shows significantly higher concentrations between about 60 hPa and 20 hPa particularly in spring. It is interesting to compare these differences with deviations between CATO and HALOE observations as presented in section 4.6.
4.6. Comparison With HALOE Observations
 In this section we are covering two topics. First, we compare CATO with HALOE to determine seasonally averaged biases between the two data sets similar to the comparison with sondes in section 4.5. Second, we investigate the variability of the differences between individual HALOE profiles and the corresponding CATO profiles sampled at the equivalent latitudes of the HALOE measurements. This serves as a measure for the ability and limitations of CATO to reproduce the full three-dimensional ozone field in the stratosphere, which is represented in the HALOE data but has to be reconstructed from a 2-D field in case of CATO. Errors associated with the step from 2-D to 3-D will therefore be referred to as 3-D reconstruction errors in the following.
4.6.1. Seasonal Biases
 In the same way as for the ozonesondes we have calculated equivalent latitudes for all HALOE profiles (see section 2.2 for details about the HALOE data used). The HALOE ozone profiles were then grouped by season and mapped onto the CATO (ϕE, θ) grid by simple averaging over all observations within a grid cell. Observations with a retrieval uncertainty larger than 20% (mainly affects observations below 100 hPa) as well as grid cells with less than 10 observations were excluded. Since HALOE has not been thoroughly validated in the troposphere yet all data below the tropopause are excluded as well.
Figure 12 shows the December-January-February (DJF) seasonal mean distributions of CATO and HALOE averaged over the 3 years 1998–2000. Note that in Figure 12 the CATO results are true seasonal means whereas the HALOE data are only averaged over a limited number of days. Relative differences in terms of 2(CATO-HALOE)/(CATO + HALOE) are presented in Figure 13. Here, CATO fields have first been interpolated onto the equivalent latitude profiles of the HALOE observations and then averaged again to the CATO grid, such that the CATO data cover exactly the same dates and locations as HALOE. Differences are generally small above 40 hPa, mostly in the range of ±10%. Below that differences are mostly within ±20% in midlatitudes but CATO values are of the order of 10 to 30% higher than HALOE in the tropics between 30°S and 30°N equivalent latitude. The largest differences occur in DJF and March-April-May (MAM) (with differences of up to 50%) and the lowest in June-July-August (JJA). In DJF this region of large positive deviations extends down to 60°S whereas in other seasons the differences in midlatitudes are mostly within ±10% and rarely exceeding 30%. The comparison with the soundings of the tropical station Hilo showed a similar picture (see Figure 10f). CATO was biased high relative to the sonde measurements between about 70 hPa and 30 hPa altitude with largest deviations occurring in spring. Note, however, that CATO values agree well with the ozonesondes below 70 hPa whereas the differences from HALOE tend to maximize around the 100 hPa level, suggesting that HALOE is also low compared to the sondes between 70 and 100 hPa. In a recent comparison between HALOE V19 and SAGE II V6.00 data, Morris et al.  found that HALOE values are lower than SAGE by 5–20% below 22 km (40 hPa) altitude and between 40°N and 40°S, which corresponds closely to the region where we find the largest differences. A low bias in HALOE observations below 22 km relative to other instruments including a balloon-borne UV-Vis spectrometer has also been reported by Borchi et al. . This suggests that compared to SAGE II, CATO would be high by only some 10–20% between 40 and 100 hPa. Wang et al.  have compared SAGE II V6.1 data with Hilo ozonesondes and found no significant bias (<10%) down to 16 km (about 100 hPa). Thus it appears that the positive bias of CATO between 30 and 70 hPa of up to 30% relative to Hilo measurements in MAM is real but that the bias between 70 and 100 hPa is considerably lower than suggested by the comparison with HALOE. Note that only a minor positive bias is seen relative to Hilo at these levels. The tropical profiles of CATO seem to be generally shifted to lower altitudes which may indicate a limitation of the reconstruction method at low latitudes where potential vorticity fields are less reliable.
 Large relative differences are also seen in September-October-November (SON) near the South Pole with CATO being lower than HALOE by up to 100%. The different vertical profiles near the South Pole (averaged over 70°S–90°S) in SON are shown in Figure 14. For comparison, the southern midlatitude profiles (40°S–60°S) of CATO and HALOE are also displayed. Near the South Pole the profiles show large differences with CATO being generally lower above 100 hPa resulting in a substantially lower TO3 column (which is essentially given by the TOMS observations). Air masses strongly depleted in ozone are seen in CATO up to about 70 hPa whereas in the HALOE profile ozone starts to increase already at 100 hPa.
 At high northern equivalent latitudes CATO agrees quite well with HALOE, mostly within ±20%. This supports our hypothesis that the large differences from the ozonesondes at Sodankylä are at least partly due to rapid ozone depletion in air masses originating from the polar vortex. Different from the balloon sondes both TOMS and HALOE can observe such high-latitude air masses only if they are advected to the south where they are exposed to sunlight.
 At the lowest θ layer the CATO reconstruction sometimes shows unrealistically high concentrations near the poles (see, for example, Figure 12a). The reason for this problem was already discussed in section 4.1.
4.6.2. The 3-D Reconstruction Errors
 The correlation between ozone VMR and PV is essential for both constructing the CATO data set in 2-D and for reconstructing the full 3-D ozone distribution from CATO. Since the CATO field at any given time is the product of a very large number of measurements, individual outliers due, for instance, to errors in the PV field only weakly affect the assimilation. For the same reason even weak correlations between PV and ozone allow us to reconstruct a realistic vertical ozone distribution in (ϕe, θ) coordinates. However, when CATO is used to reconstruct the full 3-D ozone distribution from isentropic PV and pressure fields at a given time then errors in the meteorological analyzes as well as more fundamental problems related to the breakdown of the correlation between PV and ozone due to nonconservative physical and chemical processes play a much larger role. Haynes and McIntyre  have stressed the essentially different behavior of a passive chemical tracer compared to PV under frictional and diabatic processes. Such processes as well as the rapid chemical depletion or formation of ozone tend to destroy the PV ozone correlation. Instead of using PV, Allen and Nakamura  have introduced a passive tracer in an advection-diffusion model to derive equivalent latitudes. They found that the correlation between equivalent latitudes derived from PV and from the passive tracer rapidly drops after breakup of the Arctic polar vortex and remains low throughout the summer. Their study indicates that there are clear limitations to using the PV equivalent latitude concept in certain regions and at certain times which implies that the ability of reconstructing ozone (or another passive tracer) in 3-D from a field in (ϕE, θ) coordinates is limited as well.
 In order to analyze the 3-D reconstruction errors of CATO in more detail we have computed statistics of the differences between individual HALOE profiles and CATO sampled at the equivalent latitudes of HALOE. Results are shown in Figure 15 exemplarily for autumn (SON). Figure 15b shows the pattern root mean square (RMS) difference which is the RMS after subtracting the seasonal mean values from both the HALOE and CATO data. In other words, it is the RMS after correcting for the bias between the two data sets. Results are expressed in percent of the seasonal mean concentrations, ie. of the mean of (HALOE + CATO)/2. For comparison, Figure 15a shows the standard deviation of the HALOE measurements alone as a measure for internal variability of ozone. The patterns in Figures 15a and 15b are very similar but the values in Figure 15b are typically 4–5 times lower than in Figure 15a. This suggests that a considerable amount of the observed variance of ozone at a given altitude and in a given latitude band can be successfully reproduced by CATO. This is confirmed by Figure 15c, showing the correlation (R2) between the two data sets. In the midlatitude lower stratosphere CATO reproduces between 30 and 60% of the observed variance. Surprisingly this value rises in the tropics to up to 80% at 50 hPa. Above 20 hPa the internal variability is generally low (panel a) and CATO can explain only a small part of the variance. Here the pattern RMS is probably dominated by noise in both the assimilated CATO data set and the HALOE measurements. The correlation is also very low at high southern latitudes which are influenced by air masses from the polar vortex. This supports the findings of Allen and Nakamura  that the correlation between a passive tracer and PV collapses after the breakup of the vortex. To conclude, the correlation with HALOE is very good for a large part of the stratosphere and the RMS differences are well within the range of differences between HALOE and ozonesondes [Bhatt et al., 1999] or between HALOE and SAGE II [Morris et al., 2002].
5. Conclusions and Outlook
 The present study demonstrates that it is possible to reconstruct realistic vertical distributions of a long-lived stratospheric tracer from total column observations by using information on quasi-horizontal transport diagnosed by the equivalent latitude coordinate of individual air masses. We have used a Kalman filter approach allowing us to assimilate the information provided by the total ozone observations and the meteorological analyses in an optimal way. The vertical ozone distribution was reconstructed in a framework of equivalent latitude as the horizontal coordinate and potential temperature as the vertical coordinate. Ozone concentrations change only slowly in this framework because it requires diabatic and chemical processes. This allows us to describe the evolution of ozone with the most simple dynamical model, namely with persistence. Sensitivity studies showed that the reconstructed ozone data set (CATO) is independent of the initialization and that it comes out as a robust solution to the minimization problem. However, the sensitivity tests have also demonstrated that, in contrast to the total ozone columns, the vertical distribution adapts relatively slowly to the new information introduced by the observations (the exponential decay time of the difference between two simulations starting with different initial fields is of the order of 2–3 weeks). As a consequence it is likely that the evolution of the shape of the vertical profiles lags behind the true evolution by probably less than 2–3 weeks. This problem could be resolved in the future by applying a Kalman smoother rather than a filter because a smoother would take into account not only past but also future observations. In addition, a smoother would reduce the overall level of uncertainty.
 The comparison with ozonesondes at four different stations at varying latitudes shows that the main features of seasonal and spatial variability are well represented and that also interannual variability is well captured. The comparison with observations from HALOE and the ozonesondes suggests that CATO significantly overestimates ozone concentrations between about 100 and 40 hPa in the tropics. However, the large differences from HALOE of about 30% at 100 hPa can be partly explained by a low bias in HALOE as suggested by previous studies comparing HALOE with SAGE II and other observations. CATO also shows significantly lower ozone concentrations than HALOE above 100 hPa over the South Pole during austral spring whereas no such differences are seen in the Northern Hemisphere. However, the comparison with the high-latitude ozonesonde station Sodankylä suggests that CATO is likely too low also near the North Pole during winter and spring.
 CATO is based on the carefully homogenized NIWA total ozone database of TOMS and GOME observations and is therefore suitable for the analysis of long-term trends and variability in the vertical distribution of ozone. The comparison with HALOE and ozonesondes suggests that CATO is most reliable in the midlatitude lower stratosphere whereas trend results obtained for polar and low latitudes will have to be interpreted with care. CATO is complementary to other satellite ozone data sets. It has a better temporal and spatial coverage than both SAGE and SBUV and the vertical profiles are fully consistent with total ozone columns. However, it has the drawback that the vertical distribution was inferred in an indirect way. The meteorological information required for the reconstruction was taken from the recent ERA-40 reanalysis. Since we are only using potential vorticity and pressure fields on isentropic surfaces our reconstruction does not suffer from problems that were noticed in CTM simulations and which were related to problems with the vertical wind fields of ERA-40. However, the ERA-40 data set shows unrealistic oscillations in midstratosphere to upper stratospheric temperatures which are largest near the poles and on the summer hemisphere [SPARC, 2002]. Since PV is closely related to the vertical stability of the atmosphere these problems likely also affect the distribution of PV and hence of equivalent latitudes. Some indications that this problem is also present in our data set are seen for instance in Figure 4a near the South Pole around 10 hPa.
 An extensive analysis of the CATO data set using a standard multiple linear regression model is currently underway. Preliminary results indicate that the effects of the QBO, volcanic eruptions and variations in the strength of the Brewer-Dobson circulation can readily be identified. As a potentially interesting side product we have been able to compute tropospheric column residuals at high temporal (daily) and spatial (1.25° longitude × 1° latitude) resolution between about 45°N and 45°S, providing an alternative to the many other tropospheric column products that have been established so far.
 We would like to thank MeteoSwiss and the European Centre for Medium Range Weather Forecasts (ECMWF) for granting access to the meteorological archive of ERA-40 and operational data. Samuel Oltmans (NOAA CMDL, United States) and Rigel Kivi (FMI, Finland) are acknowledged for providing very useful guidance on how to correctly use their ozonesonde measurements. Through their profound knowledge on ozone-PV relations and satellite data comparisons, two reviewers have provided important input. This work has been supported by the European Community grant through the project CANDIDOZ under contract EVK2-CT-2001-00133.