### Abstract

- Top of page
- Abstract
- 1. Introduction
- 2. Theoretical Basics of 4D-var
- 3. System Description
- 4. Variational Assimilation System Components
- 5. Data Description
- 6. Case Studies
- 7. Conclusions
- Acknowledgments
- References
- Supporting Information

[1] A novel stratospheric chemical data assimilation system has been developed and applied to Environmental Satellite Michelson Interferometer for Passive Atmospheric Sounding (ENVISAT/MIPAS) data, aiming to combine the sophistication of the four-dimensional variational (4D-var) technique with flow-dependent covariance modeling and also to improve numerical performance. The system is tailored for operational stratospheric chemistry state monitoring. The atmospheric model of the assimilation system includes a state-of-the-art stratospheric chemistry transport module along with its adjoint and the German weather service's global meteorological forecast model, providing meteorological parameters. Both models share the same grid and same advection time step, to ensure dynamic consistency without spatial and temporal interpolation errors. A notable numerical efficiency gain is obtained through an icosahedral grid. As a novel feature in stratospheric variational data assimilation a special focus was placed on an optimal spatial exploitation of satellite data by dynamic formulation of the forecast error covariance matrix, providing potential vorticity controlled anisotropic and inhomogeneous influence radii. In this first part of the study the design and numerical features of the data assimilation system is presented, along with analyses of two case studies and a posteriori validation. Assimilated data include retrievals of O_{3}, CH_{4}, N_{2}O, NO_{2}, HNO_{3}, and water vapor. The analyses are compared with independent observations provided by Stratospheric Aerosol and Gas Experiment II (SAGE II) and Halogen Occultation Experiment (HALOE) retrievals. It was found that there are marked improvements for both analyses and assimilation based forecasts when compared with control model runs without any data ingestion.

### 1. Introduction

- Top of page
- Abstract
- 1. Introduction
- 2. Theoretical Basics of 4D-var
- 3. System Description
- 4. Variational Assimilation System Components
- 5. Data Description
- 6. Case Studies
- 7. Conclusions
- Acknowledgments
- References
- Supporting Information

[2] In March 2002 the European research satellite ENVISAT was launched into a Sun-synchronous orbit, carrying the sensors Michelson Interferometer for Passive Atmospheric Sounding (MIPAS), the Global Ozone Monitoring by Occultation of Stars (GOMOS), and the Scanning Imaging Absorption Spectrometer for Atmospheric Chartography (SCIAMACHY) aboard, which are delivering an unprecedented wealth of observations of stratospheric trace gases with global coverage.

[3] It is the objective of data assimilation to provide an estimate of the state of the atmosphere from heterogeneous, irregularly distributed observational data of differencing accuracies, fused with a numerical model of the atmosphere. This is achieved mostly by the use of estimation methods adapted to large-scale problems. For the sake of mathematical rigor, an objective optimality criterion must be invoked. In most cases a Best Linear Unbiased Estimation (BLUE) is applied, which implies a least square optimum [see *Kalnay*, 2003].

[4] Sun-synchronous satellite observations are limited to measurements at a single local time, which is clearly a limitation for atmospheric chemistry. The use of spatial-temporal DA algorithms (such as 4D-Var) can effectively propagate the observation information to other times, and thus provide a complete temporal estimation of the chemical state of the atmosphere. A thorough overview of chemical data assimilation systems is provided by *Lahoz et al.* [2007] and *Geer et al.* [2006].

[5] In the realm of advanced spatiotemporal data assimilation algorithms resting on Gaussian error characteristics providing for a BLUE, there are only two families of techniques, namely the Kalman filter [e.g., *Kalman*, 1960; *Cohn*, 1997] and the 4D-var method [e.g., *Talagrand and Courtier*, 1987]. The former is a sequential method; that is, the model state is corrected at times when observations are encountered. The Kalman filter possesses the theoretical advantage that the background error covariance matrix (BECM) is evolved by a prognostic equation, and the analysis error covariances are provided by a diagnostic equation. However, with two model integrations per dimension of model-phase space, the implementation of the full Kalman filter algorithm is not feasible for atmospheric applications and complexity reduced Kalman filter algorithms must be applied [*Hanea et al.*, 2004]. To the knowledge of the authors it was for regional scale chemical data assimilation that Kalman filter implementations, which provide analysis error covariance matrices, were studied first. In the Netherlands two chemistry transport models (CTMs) were furnished with sophisticated implementations of complexity reduced Kalman filters. These include the reduced rank square root Kalman filter of the Long Term Ozone Simulation (LOTOS) model [*van Loon et al.*, 2000] and the EUROS model [*Hanea et al.*, 2004]. The reduced rank square-root approach was selected to factorize covariance matrices by a few principal components [*Verlaan and Heemink*, 1995].

[6] Unlike the Kalman filter, the 4D-var algorithm acts as a smoother, as it adjusts the initial values of the assimilating model, such that differences between observations and model state within a predefined time interval are minimized in a root-mean-square sense. The 4D-var method is sufficiently efficient to be implemented without serious simplifications. However, it lacks means to update the BECM, which must be prescribed in some way instead. While in most cases this is implemented in a static way, any dynamical evolution of the BECM must be constructed by additional information. Given typical state space vectors with dimensions of (10^{6}–10^{7}), the data volume of the squared dimension of the BECM is explicitly intractable for comprehensive three-dimensional chemistry models. As the background field is given by a short-range forecast in present assimilation systems, it is the error statistics of this forecast that is to be approximated. Further, in 4D-var there is no direct strategy to derive an analysis error estimate. Addressing the latter problem, approaches like those proposed by *Fisher and Courtier* [1995] can be applied.

[7] A first application of the 4D-var technique with a heavily reduced stratospheric reaction mechanism in connection with a trajectory model has been presented by *Fisher and Lary* [1995]. This was the first study considering the assimilation of chemically active stratospheric constituents. A state of the art tropospheric chemistry mechanism was introduced to variational assimilation by *Elbern et al.* [1997]. First extensions to the full 4D-var are given by *Elbern and Schmidt* [1999, 2001]. The 4D-var method proved flexible enough for generalization to emission rate inversion with reactive chemistry, as shown by *Elbern et al.* [2007].

[8] *Errera and Fonteyn* [2001] presented the first application of the 4D-var method in the stratosphere with a comprehensive stratospheric CTM, now the Belgian Assimilation System of Chemical Observations from ENVISAT (BASCOE; see also *Errera et al.* [2008] for further developments).

[9] Since an optimal analysis requires a realistic representation of error statistics, the treatment of the BECM constitutes a core task in designing an assimilation system. A properly specified BECM does not only balance the forecast or background error with respect to observation errors, but also guides the spreading of measurement information given a statistically well estimated influence or decorrelation length. By using known correlations, the BECM can therefore constitute a key instrument to exploit the information contents of a retrieval or observation as thoroughly as possible. In practice, the treatment of the BECM involves two independent general problems: First, an algorithmic formulation must be found to model an extremely high dimensional matrix that is generally too big to be represented explicitly. Secondly, statistically useful entries must be inferred in some suitable way.

[10] In practice, either only variances are considered (diagonal BECM [*Errera et al.*, 2008]), or the specification of covariances in chemical data assimilation mainly rests on assumptions of homogeneity (constant horizontal correlation lengths all over the globe) and isotropy (constant horizontal correlation lengths in each direction). Despite the necessary simplicity, a skillful parameterization should be capable of representing the relevant structures of the background error covariances. This includes the possibility of modeling inhomogeneous and anisotropic correlations length scales.

[11] In the realm of meteorological data assimilation, the formulation of the BECM has attracted much research (see *Bannister* [2008a, 2008b] for a comprehensive survey). An early attempt to move to anisotropic and inhomogeneous background errors is presented by *Thiebaux* [1976] by a suitably defined autoregressive scheme. *Thépaut et al.* [1996] and *Otte et al.* [2001] further demonstrated the need to relax the constraint of isotropy. A latitudinally dependent correlation function was defined by *Wu et al.* [2002] through recursive filtering. *Purser et al.* [2003] generalized the approach to variable anisotropy and inhomogeneity adaptive to geographic location. *Weaver and Courtier* [2001] applied a diffusion method, providing the same statistical properties as the recursive filtering. While for the former authors claim higher numerical efficiency, the latter show promise to better account for abrupt correlation changes, like at fronts and other air mass boundaries, due to the local control by diffusion coefficients.

[12] A straightforward approach to obtain flow dependency by anisotropy and inhomogeneity has been proposed by *Riishøjgaard* [1998], where the BECM is controlled by a function of variability of concentration levels, but also mentioned the possibility to use potential vorticity (PV) fields. While this is a direct and easy to implement method without a need to form an ensemble of model fields, the validity of the method rests on the assumption that similar field values imply similar origin, separated only by distorting flows.

[13] In the challenging domain of tropospheric chemical data assimilation, *Hölzemann et al.* [2001] introduced inhomogeneous BECMs, to account for urban-to-rural chemical regimes changes of the boundary layer. On the global scale, *Segers et al.* [2005] applied a complexity reduced global stratospheric Kalman filter system with anisotropic covariance formulation by a parameterization of correlations. Using MIMOSA (Modèle Isentrope de transport Méso échelle de Ozone Stratosphérique par Advection), a horizontally high-resolution transport model with 16 isentropic levels, it was possible to preserve fine-scale structures in the analyzed ozone field [*Fierli et al.*, 2002]. Background error correlations were flow dependent and anisotropic, specified in terms of distance and by PV field.

[14] The general objective of the first part of this study is to introduce a data assimilation system, which combines the full sophistication of a dynamically controlled BECM formulation with a complex state of the art reactive chemistry model using the 4D-var technique.

[15] The specific objectives of this paper is to validate an efficient flow-dependent formulation of the spatial background error covariances, while maintaining ability to (1) make best use of all available (satellite) data, by algorithmic capability to extend observation results spatially, while preserving the BLUE property, (2) ensure chemical and dynamical consistency by application of a state of the art chemistry mechanism, and (3) to provide numerical efficiency (grid design, parallelization), to allow for near real time operation. The system SACADA (Synoptic Analysis of Chemical Constituents by Advanced Data Assimilation), presented in this study, has been designed efficiently enough to provide for daily routine operations.

[16] In order to comply with the first item, the study is designed to implement and test the diffusion approach proposed by *Weaver and Courtier* [2001]. The resulting background error covariance operator will be shown to be well suited for the application with large models and allows for anisotropic and inhomogeneous background error correlations, a feature that was utilized to devise a flow-dependent formulation of the BECM.

[17] This work lays the foundation for a follow-up study addressing a more sophisticated a posteriori validation of the assimilation results in observation space [*Desroziers et al.*, 2005]. This paper is organized as follows. Section 2 describes the theoretical background of the assimilation approach used. In section 3 the meteorological driver model and its geodetic grid, together with the chemistry transport model are presented. Further, the data assimilation setup is given in section 4, with emphasis placed on background error covariance modeling. The satellite data involved in this study are introduced in section 5. Results and statistical evaluations of the assimilation runs are presented in section 6, followed by conclusions made in section 7.

### 2. Theoretical Basics of 4D-var

- Top of page
- Abstract
- 1. Introduction
- 2. Theoretical Basics of 4D-var
- 3. System Description
- 4. Variational Assimilation System Components
- 5. Data Description
- 6. Case Studies
- 7. Conclusions
- Acknowledgments
- References
- Supporting Information

[19] Let _{i} denote the nonlinear model operator that propagates the atmospheric state **x**_{0} at time *t*_{0} to time *t*_{i}. Further, it is assumed that the model and its controlling parameters, except initial values, are of sufficient quality for a forecast of the time span of interest, with statistics of forecast errors known in terms of its error covariance matrix. In addition, given a set of observations or satellite retrievals over a time interval or assimilation window [*t*_{0}, *t*_{N}], we seek for a BLUE of the model's *n* initial values of the state variable **x**_{0} = **x**(*t*_{0}). It is assumed that background and observation errors are normally distributed and uncorrelated.

[20] The four-dimensional variational method minimizes the cost function:

Adopting the *Ide et al.* [1997] notation as far as possible, **y**_{i} is the vector of observations available within time step *i*, **x**^{b} is the first guess or background state at initial time, typically obtained from an earlier model forecast. The initial chemical state **x**_{0} of the model at time step *i* = 0, at the beginning of the assimilation window, is the optimization parameter, where the most probable state is to be identified. The background error and observation error covariance matrices are denoted by **B** and **R**, respectively. The observation operator *H* maps the model state onto the observation state. Superscript *T* denotes transposition. As is obvious from (1), *J*^{b} are the partial costs resulting from the deviation of initial state **x**_{0} from the background state **x**^{b}, while *J*^{o} gives the partial costs arising from the deviation of the observations **y**_{i} from the model equivalents *H*(_{i}(**x**_{0})). Equation (1) implements the model as a strong constraint.

[21] Minimization algorithms, which are suitable for high-dimensional problems, include quasi-Newton or Conjugate-Gradient methods and require gradients of the cost function with respect to the control variables **x**_{0}. The calculation of *J*^{o} is the computationally most demanding task of 4D-var data assimilation, given the number of control variables in atmospheric models is on the order 10^{6}–10^{7}. The feasible strategy to accomplish the calculation of the gradient makes use of the adjoint model operator. The gradient of *J* with respect to the initial model values **x**_{0} is given by [*Talagrand*, 1997]

[22] Here, **M**_{i}* is the adjoint model operator, linearized backward in time from time step *N* until time step *i*, while **H** is the linearized observation operator *H*.

### 7. Conclusions

- Top of page
- Abstract
- 1. Introduction
- 2. Theoretical Basics of 4D-var
- 3. System Description
- 4. Variational Assimilation System Components
- 5. Data Description
- 6. Case Studies
- 7. Conclusions
- Acknowledgments
- References
- Supporting Information

[90] The novel SACADA 4D-var system for operational assimilation of stratospheric observations has been developed ab initio, aiming to provide a middle atmospheric chemistry analysis tool, which is efficient in terms of processing available retrieval information and use of computational resources. The common grid structure and the common time steps of the meteorological driver module GME with the chemistry data assimilation section avoids spatial and temporal interpolation of the meteorological fields with associated information loss and error generation. In particular, a consistent representation of vertical wind fields is available for the solution of the advection-reaction equation.

[91] Moreover, the numerical efficiency benefits considerably from the icosahedral grid, where the computational costs are reduced by about 30%, compared to CTMs employing a traditional latitude-longitude grid. In addition, the semi-Lagrange horizontal transport scheme applicable for short simulation intervals, leads to an excellent efficency of the new system, where the number of transported constituents is large. While for adjoints of semi-Lagrangian schemes on traditional grids considerable efforts are required to maintain accuracy at the poles, the icosahedral grid is not affected. The efficient system design enables the application of the computationally costly 4D-var technique some eight times faster than real time, and opens options for further grid refinements and chemical mechanism extensions.

[92] Particular efforts were devoted to implement spatial correlations with the BECM. This problem was solved by introducing the diffusion approach following *Weaver and Courtier* [2001]. A modified potential vorticity was adopted to identify air mass structures, allowing to estimate anisotropic and inhomogeneous correlation lengths accordingly. The implementation presented in this work assumes larger background error correlations along isopleths of potential vorticity in regions where large gradients of potential vorticity prevail. At the polar vortex edge, analyses of chemical constituents appear to be more consistent with the dynamics than those made with a homogeneous and isotropic formulation, which ignores dynamic patterns.

[93] The diffusion approach proved to have several advantages, as it (1) saves the storage of a full BECM, which is replaced by an operator with the same statistical properties, (2) does not need the assumption of separability of horizontal and vertical correlations, as related diffusion operators are alternatingly applied during the three-dimensional integration, (3) allows for an easy preconditioning of the minimization problem by straightforward calculation of the square root of the BECM, (4) is numerically efficient, as the complexity of the calculation is linear with the dimension of the model grid, (5) is amenable for flow-dependent design of spatial covariances, allowing for pronounced variations of the correlation lengths, and (6) can be easily adapted to nonstandard grid design as the icosahedral grid, without any difficulties.

[94] A suite of two case studies, comprising 1 September to 15 October 2002, and 21 October to 30 November 2003 served for validation of the assimilation system. EnviSat-MIPAS data products have been assimilated and the assimilation results have been validated with independent (not assimilated) data from SAGE II and HALOE. A sequence of tests proved the quality of the assimilation results. The a posteriori validation of normalized costs demonstrated a fast convergence toward the optimal value *J*_{p}^{a} ≈ 1/2 after 4 days of spin-up. Display of (O − A) and (O − B) density distributions reveal nearly bias free analyses with significantly reduced variance of PDF. The PV controlled dynamical BECM formulation exhibits advantages in areas with pronounced PV gradients. Comparisons of MIPAS based assimilation results with not assimilated SAGE II and HALOE retrievals revealed significantly improved analyses, albeit in the limits of agreements between the infrared sounder and the occultation sensors. It could be shown that the application of the assimilation system also improved short-range forecasts, as demonstrated by the background based model runs.

[95] In a companion paper (J. Schwinger and H. Elbern, Chemical state estimation for the middle atmosphere by 4-dimensional variational data assimilation: 2. A posteriori validation of error statistics in observation space, submitted to *Journal of Geophysical Research*, 2010), an in-depth a posteriori validation is provided, where the mutual consistency of the background and observation error covariance matrices will be established and an analysis error assessment in observation space will be presented.

[96] The SACADA system is running an operational service providing daily analyses in near real time.

### Acknowledgments

- Top of page
- Abstract
- 1. Introduction
- 2. Theoretical Basics of 4D-var
- 3. System Description
- 4. Variational Assimilation System Components
- 5. Data Description
- 6. Case Studies
- 7. Conclusions
- Acknowledgments
- References
- Supporting Information

[97] The authors are highly indebted to the German Weather Service and D. Majewski for giving access to GME code and providing advice. W. Joppich, S. Pott, and H.-G. Reschke, SCAI, Fraunhofer Society, gave a lot of support in adapting the vertical grid structure of GME to the needs of stratospheric modeling. J. Hendricks, DLR, provided advice on the use of the chemical mechanism including heterogenous chemistry to the SACADA system. We are very grateful to D. Poppe, ICG-2, Research Centre Jülich, and E.-P. Röth, University Essen and ICG-2, FZ Jülich, for a critical final review of the extended version of the chemistry mechanism, and to Anne Smith, NCAR, for provision of photolysis rates. MIPAS data have been processed and provided by ESA. We are grateful to G. Brasseur, NCAR, and A. Sandu, Virginia Tech, for giving access to SOCRATES and KPP software, respectively. SAGE II data were obtained from NASA Langley Research Centre, and HALOE data were obtained from Hampton University, Virginia, and NASA Langley Research Center. We are further indebted to the SACADA team, most notably M. Riese and L. Hoffmann, ICG 1, FZ Jülich, T. von Clarmann, IMK, KIT, and H. Bovensmann, IFE, University of Bremen, for manifold discussions on satellite retrieval error characteristics. The meteorological data for driving GME were obtained from the European Centre for Medium-Range Weather Forecasts (ECMWF). Computational resources were provided by University of Cologne's computer centre RRZK and the Jülich Supercomputing Centre. This work was funded by the German Federal Ministry of Education and Research in the frame of the funding program AFO 2000 with the grant FZK 07ATF48. The authors want to thank three anonymous reviewers, who helped to improve the manuscript.