Depending on the altitude, ozone in the Earth's atmosphere has different effects. In the stratosphere, ozone filters the harmful ultraviolet part from the incoming solar radiation, preventing it from reaching the surface. Near to the surface, ozone is a pollutant, which has negative effects on human health and can reduce crop yields. At the same time, ozone is a greenhouse gas with an important role in the temperature of the atmosphere.

Because of the important role ozone has in climate change, it has been designated as an essential climate variable (ECV) by the Global Climate Observing System (GCOS) of the World Meteorological Organization . In the GCOS report, it is stressed that the full three-dimensional distribution of ozone is required.

The European Space Agency (ESA) has initiated the Climate Change Initiative (CCI) programme, which aims at long-term time series of satellite observations of the ECVs (http://cci.esa.int/). One of the sub-programmes is the Ozone CCI project (http://www.esa-ozone-cci.org/) that focuses on homogenized datasets of total ozone from different sensors , stratospheric ozone distribution from limb and occultation observations e.g., and the vertical ozone distribution from nadir observations e.g.. Long-term ozone datasets were also produced by the European Centre for Medium-Range Weather Forecasts (ECMWF) reanalyses such as ERA-40 and its successor ERA-Interim . Although primarily intended for improvement of the weather forecast, the assimilation of ozone is an integral part of theses reanalyses. ERA-40 is described in more detail in and ERA-Interim in . Total ozone column measurements from different satellite instruments were assimilated into a chemical transport model for the multi-sensor reanalysis (MSR) of ozone , spanning a 42-year period between 1970 and 2012.

Vertical ozone measurements from space-based ultraviolet (UV) instruments started with the Solar Backscatter Ultraviolet (SBUV) instruments from 1970 onwards on different satellites e.g.. Later, satellite instruments with higher resolution and increased spectral coverage were launched, for example Global Ozone Monitoring Experiment (GOME) onboard ERS-2 in 1995 , SCanning Imaging Absorption spectroMeter for Atmospheric CHartographY (SCIAMACHY) onboard Envisat in 2002 , Ozone Monitoring Instrument (OMI) onboard Aura in 2004 and Global Ozone Monitoring Experiment 2 (GOME-2) onboard Metop-A/B in 2006/2012 . Each location on Earth is typically observed once or twice a day by these satellites, so it is not possible to get global coverage at a specific time of the day. The retrieved ozone profiles from ultraviolet-visible (UV-VIS) nadir observations have a limited vertical resolution due to the smoothing effect in the retrieval e.g.. The vertical resolution varies between 7 and 15 km see. To derive gridded 3-D ozone distributions at fixed time intervals we use data assimilation, which combines the information present in the model and the observations, giving the optimal estimate of the ozone concentration. Either the retrieved ozone data or the radiance data from the instrument can be assimilated into the model. showed that both methods are equivalent. However, assimilating retrieved ozone data considerably simplifies the observation operator and reduces the number of measurements to assimilate. Since the measurement, averaging kernel (AK) and error covariance matrices are all used in our assimilation algorithm, all information gained from the retrieval is also present in the resulting assimilated model fields.

Two commonly used types of data assimilation are 4DVAR and (ensemble) Kalman filtering. For example, ozone profiles and total columns from different instruments (such as GOME) were assimilated using a 4DVAR assimilation scheme in the production of the ECMWF ERA-Interim reanalysis see. The Belgian Assimilation System for Chemical ObsErvations (BASCOE, http://bascoe.oma.be/; ) is a stratospheric 4DVAR data assimilation system for multiple chemical species including ozone and nitrogen dioxide. BASCOE is used in the Monitoring Atmospheric Composition and Climate (MACC) and Copernicus Atmosphere Monitoring Service (CAMS) projects for atmospheric services, the stratospheric ozone analyses from the MACC project are evaluated in . Recently, BASCOE has been coupled to the Integrated Forecast System of the ECMWF . 4DVAR is well suited to assimilate large amounts of observations, and the analysis provides a smooth field at the time of the assimilation. However, there are two disadvantages of 4DVAR with respect to Kalman filter techniques. First, 4DVAR requires the development and maintenance of an adjoint model, which is a complicated process. Second, 4DVAR does not produce a direct estimate of the uncertainty in the ozone field, although such an estimate can be derived using computationally expensive techniques.

The model covariance matrix is an integral and essential part of a Kalman filter, but it is difficult to derive and computationally expensive in the analysis calculation. Therefore, most Kalman filter implementations try to approximate the model covariance matrix. In the ensemble Kalman filter, a selection of the ensemble members can be used to approximate the model covariance see. used an ensemble Kalman filter to assimilate different trace gas measurements from multiple satellite instruments into a chemical transport model.

In this research, we follow the Kalman filter approach described in , where the model covariance matrix is parameterized into a time-dependent standard deviation field and a time independent correlation field. The algorithm was updated and used by to subtract the assimilated stratospheric ozone column from the total column in order to obtain the tropospheric ozone column. We have implemented several major updates and improvements in the algorithm compared to the version of . We check the observational error characterization, redefine the model error growth and derive a new correlation matrix for the ozone field. The new algorithm is the first that simultaneously assimilates nadir ozone profiles from multiple high-spectral-resolution satellite instruments. We demonstrate the new algorithm by assimilating ozone profile observations from GOME-2 and OMI for the period 2008–2011 into the chemical transport model TM5 e.g.. To minimize the bias between the two instruments, we developed a bias correction based on ozone sondes to be applied to the observations before assimilation. A bias correction based on total column measurements from ground stations was earlier used for the MSR of total ozone . Since we assimilate ozone profiles, we require an altitude dependent bias correction for which ozone soundings are selected.

In Sect.  we briefly describe the ozone profile observations, and in Sect.  the chemical transport model that we use for the data assimilation is described. Section  gives a short overview of the assimilation algorithm, Sect.  describes the improvements applied to the assimilation algorithm and the results will be shown in Sect. . In Sect.  we demonstrate the performance of the assimilation algorithm over the Tibetan Plateau. A discussion of the results is given in Sect.  and the conclusions are presented in Sect. .

Data from the UV-VIS satellite instruments GOME-2 and OMI are available for the last 10 years.

GOME-2 was launched onboard Metop-A in 2006. The instrument measures the solar light reflected by the Earth's atmosphere between 250 and 790 nm. For the retrievals used in this research, the radiance measurements are binned in the cross-track and along-track directions such that the ground pixels measure approximately 160km×160 km. The ozone profiles for GOME-2 are retrieved with the OPERA retrieval algorithm, which is described in . We increased the number of layers in this study from 16 to 32 for more accurate radiative transfer model results.

OMI was launched onboard Aura in 2004. The instrument measures the solar light reflected by the Earth's atmosphere between 270 and 500 nm. One important difference between OMI and GOME-2 is that OMI does not use a scanning mirror like GOME-2, but a fixed 2-D CCD detector. One dimension of the detector is used to cover the spectral range, the other is used to cover the cross-track direction. The ground pixels for the profiles retrieved from the UV-VIS spectrum measure approximately 13km×48 km in nadir. Note that only 1 in 5 scan lines are retrieved. The size of the ground pixels is increasing towards the edge of the swath. OMI has two UV channels that are used in ozone profile retrieval: UV1 and UV2. UV1 has 30 cross-track pixels, while UV2 has 60 cross-track pixels. The UV2 pixels are therefore averaged to coincide with the UV1 pixels. A description of the OMI ozone retrieval algorithm and validation results with respect to ground measurements and other satellite instruments can be found in .

The algorithms used to retrieve the ozone profiles from GOME-2 and OMI are both based on an optimal estimation technique. The state of the atmosphere is given by the state vector x, while the measurement is given by the measurement vector y and error ϵ. These two vectors are related by the forward model F according to y=F(x)+ϵ. Following the maximum a posteriori approach , the solution is given by x^=xa+Axt-xa+Gϵ,S^=I-ASa,A=GK=SaKTKSaKT+Sϵ-1K, where x^ is the retrieved state vector, xa is the a priori state of the atmosphere, A is the averaging kernel, xt is the “true” state of the atmosphere, G is the gain matrix (SaKTKSaKT+Sϵ-1), Gϵ the retrieval noise, S^ is the retrieved covariance matrix, I is the identity matrix, Sa is the a priori covariance matrix, K is the weighting function matrix or Jacobian (it gives the sensitivity of the forward model to the state vector) and Sϵ is the measurement covariance matrix.

The averaging kernel can also be written as A=∂x^/∂xt and gives the sensitivity of the retrieval to the true state of the atmosphere. The trace of A gives the degrees of freedom for the signal (DFS). For the cloud-free retrievals over the ozone sonde stations used in this study, the mean DFS for GOME-2 is 5.0 and for OMI is 5.1. When the DFS is high, the retrieval has learned more from the measurement than in the case of a low DFS, when most of the information in the retrieval will depend on the a priori state. The total DFS can be regarded as the total number of independent pieces of information in the retrieved profile. The rows of A indicate how the true profile is smoothed out over the layers in the retrieval and are therefore also called smoothing functions. Ideally, the smoothing functions peak at the corresponding level and the half-width is a measure for the vertical resolution of the retrieval.

Because the sensitivity of the retrieval to the vertical ozone distribution is represented by the averaging kernel, it is important to include the averaging kernel in the assimilation algorithm. Together, the retrieved state vector, the averaging kernel and error covariance matrices represent all information gained from the retrieval .

The model used in the assimilation is a global chemistry transport model called TM5 (Tracer Model, version 5), see for an extended description. The (tropospheric) chemistry of TM5 has been evaluated in and included into the integrated forecasting system of the ECMWF .

In the current model setup used for the assimilation of the ozone profiles, TM5 runs globally with grid cells of 3∘ longitude × 2∘ latitude, on 45 pressure levels. The pressure levels are a subset of the 91-level pressure grid from the ECMWF. The meteorological data used to drive the TM5 tracer transport are taken from the ECMWF operational analysis fields, produced on these 91 pressure levels.

Above 230 hPa, ozone chemistry is parameterized according to the equations described by , using the parameters of version 2.1. In the troposphere, the ozone concentrations are nudged towards the Fortuin and Kelder climatology , with a relaxation time that increases from 0 days at 230 hPa to 14 days at 500 hPa and lower. No other trace gasses are modelled in this setup, which makes this version of TM5 fast and computationally cheap. Ozone concentrations are prevented from following the model equilibrium state too closely by the frequent confrontation of the model with the observations during the assimilation process.

The assimilation algorithm uses a Kalman filter, and is described in . The state vector xi (i.e. the ozone distribution at time t=i) and the measurement vector yi (i.e. the retrieved profiles at time t=i) are given by xi+1=Mxi+wi,wi∼N0,Qi,yi=Hxi+vi,vi∼N0,Ri, where M is the model that propagates the state vector in time. It has an associated uncertainty w, which is assumed to be normally distributed with zero mean and covariance matrix Q. The observation operator H, which includes the averaging kernel, gives the relation between x and y. The uncertainty in y is given by v, which is also assumed to have zero mean and covariance matrix R (which is identical to S^ in the retrieval equations). In matrix notation, the propagation of the state vector and its covariance matrix (P) are given by xi+1f=Mxia,Pi+1f=MPiaMT+Qi, where xf and xa are the forecast and analysis state vectors, respectively, at time t=i, i.e. before and after assimilation of the observations. The observations are assimilated according to xia=xif+Kiyi-Hxif,Pia=I-KiHiPif,Ki=PifHiTHiPifHiT+Ri-1, where K is called the Kalman gain matrix, and the matrix H is the sensitivity of the observation operator with respect to the state.

The observation operator interpolates the model field to the observation location, converts the model units to the retrieval units and takes the smoothing of the satellite instruments into account by incorporating the averaging kernel. The model grid cells are 3∘×2∘ (longitude × latitude), much larger than the satellite ground pixels and therefore no horizontal interpolation is needed. The model profile, expressed DUlayer-1, is converted to the pressure levels of the retrieval grid by applying a simple linear interpolation in the 10log (hPa) domain. For example, if the L2 profile layer overlaps with three model layers for 20, 100 and 30 %, the interpolated model partial column is 0.2×DU1+1.0×DU2+0.3×DU3 (where DUi is the partial column for layer i). Finally, the observation operator H is formed by applying the averaging kernel A to the difference between the state vector x and the a priori profile ya used in the retrieval: Hx=ABCx-ya, with C being the unit conversion (from the models kg grid-cell-1 to the observations DUlayer-1) and B being the vertical interpolation. The sensitivity matrix H used in Eqs. () and () is constructed as H=ABC.

In general, the number of elements in an ozone profile is much larger than the degrees of freedom (about 5 to 6). We can therefore reduce the number of data points per profile by taking the singular value decomposition of the A, and only retain the vectors with a singular value larger than 0.1 (this is an absolute threshold and not relative to the maximum singular value). The profiles and matrices are transformed accordingly.

The computational cost of the assimilation algorithm can be further reduced by minimizing the size of the model covariance matrix P. The TM5 model runs in the current setup on a horizontal grid of 2∘×3∘ (latitude × longitude) on 44 layers, which makes the size of the covariance matrix (475200)2 elements. A number of different approaches exist to minimize the size of the model covariance matrix. For example, in , the number of dimensions is reduced by only assimilating total columns, while the horizontal correlation depended only on the distance between the model grid cells. Here, we follow the approach described by , by parameterizing the model covariance into a time-dependent standard deviation field and a constant correlation field. At each time step, the model's advection operator is applied to the standard deviation field. The error growth (i.e. the addition of Q in Eq. ) is modelled by a simple mathematical function, more details can be found in Sect. . The model covariance matrix can now be calculated according to P=DσCDσ, with Dσ being a matrix with the values of the standard deviation σ on the diagonal and C the correlation matrix. The correlation matrix is calculated differently than in , more details can be found in Sect. .

Unfortunately, the HiPifHiT+Ri matrix in the Kalman filter (Eq. ) is badly conditioned, which makes the inversion sensitive to noise. Therefore, the eigenvalue decomposition of this matrix is calculated and the measurements are projected on the largest eigenvalues, which in total represent 98 % of the original trace of the matrix.

For the numerical stability of the assimilation algorithm, the difference between the observation and the model should not be too large. A filter is implemented that rejects the observation when the absolute difference between the observation and the model forecast is larger than 3 times the square root of the sum of the variance in the observation and the variance in the forecast: absyi-Hxif≥3σyi2+σxif2, with σy and σxf the standard deviation of the observation y and the forecast H(xf) for layer i, respectively. Note that this is done on a layer-by-layer basis, i.e. if in one layer the difference is too large, the whole observed profile is discarded.

Not all available ozone profiles can be assimilated into TM5 because the computational cost would be too high. Averaging retrievals on the model grid (sometimes called superobservations) was not possible because the assimilation algorithm described in this paper requires AKs and averaging AKs is not straightforward. Therefore 1 out of 3 GOME-2 profiles and 1 out of 31 OMI profiles are used. These numbers are chosen such that more or less the same number of observations are assimilated for each instrument, taking into account the decrease in available pixels due to the row anomaly in OMI. For OMI, the outermost pixels on each side of the swath are neglected, because of the large area of these pixels. Of the resulting retrievals, only cloud-free scenes (cloud fraction ≤0.2) are assimilated in order to get the maximum amount of information from the troposphere.

The first version of our assimilation algorithm was described in . They assimilated GOME ozone profiles for the year 2000. This dataset was extended to the period 1996–2001 by who derived tropospheric ozone for this period. The assimilated GOME observations in the previous algorithm version had a pixel size of 960km×100 km, much larger than the pixels in the current research. Since 2009, the assimilation algorithm has been further developed and improved for use with GOME-2. The improved resolution of GOME-2 and OMI ozone profiles and improved retrievals offer new possibilities, but also require adaptations in the data assimilation. It is the first time that ozone profiles from two nadir looking instruments, GOME-2 and OMI, are assimilated simultaneously. This has resulted in a significant number of updates and improvements to the assimilation algorithm compared to the version described in and , which are outlined in the following sections.

We have assimilated GOME-2 (on Metop-A) and OMI ozone profiles for a period of 4 years between 2008 and 2011 using the Kalman filter algorithm described in the previous sections. In total, four model runs were performed: a “free” model run without assimilation, a model run with assimilation of GOME-2 ozone profiles only, a model run with assimilation of OMI ozone profiles only and a model run with simultaneous assimilation of GOME-2 and OMI ozone profiles.

To demonstrate the performance of the assimilation algorithm we analysed the results for a day above the Tibetan Plateau (located between 30 and 40∘ N), where a highly dynamical atmosphere exists. This makes it an interesting area to study atmospheric dynamics, and difficult for modelling so it can serve as a test case to see if the dynamics in the model are correctly implemented. On 25 February 2008 a stratosphere–troposphere exchange event was observed in GOME-2 data , which can also be observed in the assimilation output. In Fig. , ozone concentrations from the ERA-Interim reanalysis are plotted as contours over the ozone concentrations from the model runs with and without simultaneous assimilation of GOME-2 and OMI. There is a significantly better agreement between the two datasets north of 35∘ N at pressure levels between 70 and 10 hPa. Even though the GOME-2 and OMI instruments have limited sensitivity in the troposphere, the tropospheric ozone concentrations of the ERA-Interim reanalysis and assimilated tropospheric ozone are in better agreement north of the Tibetan Plateau. There are also two stratosphere–troposphere exchanges (STE) visible, at 30 and 60∘ N. These STEs are associated with strong jet-streams (perpendicular to the page) reaching wind speeds of up to 50 ms-1 at 250 hPa.

When two instruments are assimilated simultaneously, their differences should be taken into account. For example, the algorithms used for the retrieval of GOME-2 and OMI ozone profiles both produce partial columns. However, the number of layers in the retrievals differ and the sensitivity of the retrieval is expressed by the averaging kernel. Both the different vertical resolution and the averaging kernel are incorporated into the observation operator H. Both instruments have different horizontal resolution, something which has not been taken into account in the current version of the assimilation algorithm. The measurement principle of GOME-2 (i.e. a cross-track scanning mirror) is different than that of OMI (i.e. a fixed 2-D CCD detector). As a result, the ground pixel size of GOME-2 is constant, while that of OMI varies across the track. Therefore, the representation error of OMI will increase towards the edges of the swath. The effect of the changing OMI footprint size has not been investigated. To get an idea of the sub-grid-cell variation in the ozone concentration, we performed a small experiment where we assimilated the same observations (i.e. GOME-2 and OMI) into TM5 running on a 1∘×1∘ grid (as opposed to the standard 3∘×2∘ used in this article). The total column standard deviation of the six 1∘×1∘ grid cells covered by a single 3∘×2∘ grid cell is much smaller than the error on the total column. Therefore, the representation error due to the large grid cells is not significant. A more thorough check on the instruments behaviour throughout time might have revealed the effect of the OMI L0 to L1b processor update sooner. The threshold of the parameter in the OmF filter might be made instrument and time dependent in order to minimize the effect on the number of assimilated pixels.

Two different instruments can be biased with respect to each other. In order to minimize the bias, a bias correction scheme has been implemented with respect to ozone sondes. We used cloud-free observations (max. cloud fraction 0.20) for the bias correction in order to get a maximum amount of information from the troposphere. As a consequence, we could not use all available sondes in deriving the bias correction. Sudden changes in the bias correction parameters are visible at the start of the year, when the parameters are changed. To minimize these changes, it might be worthwhile to implement an interpolation scheme for the bias correction parameters similar as for the MSR data see.

The model can run a full chemistry scheme, but instead a parameterized chemistry scheme has been used in favour of speed. Another possibility to increase the accuracy of the model is to increase the horizontal resolution from the current 3∘×2∘ (long. × lat.) to 1∘×1∘ for example. However, in both cases it might be necessary to reduce the vertical resolution of the model to keep the computational cost at an acceptable level.

The model covariance matrix is also an expensive step in the assimilation algorithm. We have reduced the calculation cost by parameterizing it into a time-dependent error field and a time-independent correlation field. The data from April 2008 was used to derive the correlations, which were then used for the whole assimilation period. The assumption that the derived correlations are constant throughout time has not been tested.

An algorithm for the simultaneous assimilation of GOME-2 and OMI ozone profiles has been described. The algorithm uses a Kalman filter to assimilate the ozone profiles into the TM5 chemical transport model. Compared to previous versions, the algorithm is significantly updated. The observational error has been characterized using a newly developed in-flight calibration method. Since the Kalman filter equations are too expensive to calculate directly for the current setup, the model covariance matrix is divided into a time-dependent error field and a time independent correlation field. The time evolution is applied to the error field only, while the correlation is assumed to be constant. The model error growth is modelled by a new function that prevents the error from increasing indefinitely, and the correlation field has been newly derived using the NMC method. Large biases between retrievals of the two instruments might destabilize the assimilation. To avoid this, a bias correction using global ozone sonde observations has been applied to the retrieved ozone profiles before assimilation.

Four model runs were performed spanning the years between 2008 and 2011: without assimilation, with assimilation of GOME-2 only, with assimilation of OMI profiles only and with simultaneous assimilation of both GOME-2 and OMI profiles. Depending on the altitude, the OmF and OmA for one instrument might be larger than the other, which might change in the course of time. Assimilating the observations from these instruments simultaneously increases the overall sensitivity of the assimilation. Two notable instrumental effects are the band 1A/1B wavelength shift for GOME-2, which causes a significant decrease in OmF and OmA. For OMI, after the L0 to L1b processor update on 1 February 2010, the uncertainty in the observations is too small with respect to the method of in-flight validation of the uncertainties presented in this paper. This caused a decrease in the number of assimilated observations for both GOME-2 and OMI. The expected and observed OmF and OmA are more similar for the combined assimilation than for the separate assimilations. Validation with sondes from the WOUDC shows that the combined assimilation performs better than the single sensor assimilation in the region between 100 and 10 hPa where GOME-2 and OMI are most sensitive. The ozone concentrations in the troposphere are also affected by the assimilation, even though the instruments have limited sensitivity in that region. The biases of the assimilated ozone fields are smaller than those of the observations. The assimilated ozone fields are produced at regular time intervals and have no missing data. Despite the limited vertical resolution of GOME-2 and OMI, a case study of an STE over the Tibetan Plateau shows that the assimilation of ozone profiles can improve the ozone distribution in a highly dynamical region.