a. 4D-Var in terms of increments

4D-Var aims at providing the best estimate x^a of the unknown true state x^t at t₀, the initial time of the assimilation window. This estimate, referred to as analysis, is a particular initial condition of the propagation (or forecast) model . It is computed using the observations y_o available over the time interval [t₀, t_P], where t_P is the final time of the assimilation window. The model solution x(t_i) at the observation time t_i is compared to the observations y_o(t_i) via the observation operator _i. The 4D-Var cost function is

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where the background (or prior estimate) field x_b is an estimate of the true state with an error derived from the realization of an unbiased random variable of covariance 𝗕₀. The instrumental and representativeness observation errors are included in the covariance matrices 𝗥_i (Lorenc 1986). The model state x(t_i) at the observation time t_i is the propagation of the initial state x(t₀) via the operator _i:

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Courtier et al. (1994) proposed a formulation of the 4D-Var cost function in terms of increments. The increment δx(t₀) is the difference between the initial and the background states, δx(t₀) = x(t₀) − x_b. Their formulation assumes that the tangent linear operators 𝗠_i and 𝗛_i of _i and _i, respectively, satisfy

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for acceptable magnitude of the perturbations δx(t₀) and δx(t_i). Under the linearity hypothesis whose validity is discussed in Tremolet (2004), Eqs. (3) and (4) are exact. When the assimilation operators are at most weakly nonlinear during the assimilation window, assuming Eqs. (3) and (4) are suitable approximations, one can express the model equivalent of the observations as a function of the following increment:

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Under this linear approximation, the nonlinear cost function of Eq. (1) is approximated by the linear cost function:

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Here δy(t_i) = 𝗛_i𝗠_iδx(t₀) and d(t_i) is the innovation vector at observation time t_i, which represents the distance between the observations and the background state propagated at t_i with the nonlinear model, and projected in the observation space with the nonlinear observation operator:

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The solution of the assimilation problem is obtained by computing the increment δx_a(t₀) that minimizes the cost function of Eq. (6). The analysis field x_a(t₀) is the sum of this increment and the background state, x_a(t₀) = x_b + δx_a(t₀). In practical applications, the minimum of Eq. (6) is computed by a minimization algorithm that generally requires the gradient of the cost function. The gradient is usually computed using adjoint methods. Nevertheless, the minimum of Eq. (6) can be expressed analytically by looking at the solution that nullifies the gradient. One can demonstrate that the solution is

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with

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where y_o is the generalized observation vector that contains all the observations available over the assimilation interval; 𝗕 and 𝗥 are the generalized covariance matrices of background and observation errors, respectively. In the same way, and are the generalization of _i and _i, with their tangent linear form 𝗠 and 𝗛, respectively. Equations (8) and (9) establish the link between the linear form of 4D-Var in terms of increments and the statistical estimation approach, as it also represents the analysis step of the extended Kalman filter (Maybeck 1979).

b. The outer loop

The 4D-Var in terms of increments requires us to linearize the assimilation operators and . The first time, the linearization is computed in the vicinity of the background. It leads to the quadratic cost function of Eq. (6). The minimization of this function results in a first analysis. In a second step, this analysis can be used as a new guess in the vicinity of which to linearize the assimilation operators. This leads to a new cost function to minimize, which results in a new analysis. This iterative process is commonly denoted the “outer loop.” Its aim is to approach the minimum of the nonlinear 4D-Var cost function with a succession of quadratic cost functions minimizations.

For the outer-loop formulation, the solution xⁿ(t₀) of the iteration n is searched as an increment δxⁿ(t₀) to be applied to the analysis x_aⁿ⁻¹(t₀) resulting from the previous iteration:

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The analysis of Eq. (10) is then propagated in time by the nonlinear forecast model in order to be compared to the observations. In terms of increments, the model equivalent of the observations is expressed as

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where 𝗛_iⁿ and 𝗠_iⁿ denote, respectively, the linear observation operator and the linear model used for the iteration n. The linear model can differ from the tangent linear of and can differ from one iteration to another. This is the basis of incremental variational methods. In practical applications, 𝗠ⁿ is the tangent linear of ⁿ where ⁿ is a version of , which can have a coarser horizontal resolution and could reflect a less detailed description of the physical laws.

At each iteration n, observations are compared to the best current estimate of the true state. This is another key point of the outer-loop process. Actually, at iteration n, observations are compared to the first guess x_aⁿ⁻¹. This current guess is the best estimate of the true state given the linear estimation of the previous iteration n − 1; thus, it is updated at each iteration. The innovation vector at time t_i is then the difference between the observations and the model equivalent of the state x_aⁿ⁻¹ given by Eq. (11):

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As previously described and as explained by Courtier et al. (1994), the outer loop was initially used to reduce the cost of the assimilation process. But the outer loop presents other interesting characteristics. It can be seen as a way to iteratively solve a nonlinear problem by a succession of (simpler) linear problems: successive linearizations of the assimilation operators are recomputed around an updated trajectory that gets closer to the truth at every iteration. The current analysis increment becomes progressively smaller and therefore compliant with the linearity hypothesis. For these reasons, we aim at using the outer loop in conjunction with a cheaper method than 4D-Var.

c. 3D-FGAT with the outer loop

In the incremental approach proposed by Courtier et al. (1994), the linear model 𝗠 can differ from the tangent linear of . The FGAT approach represents an extreme case where the linear model 𝗠 is replaced by the identity matrix in Eq. (5), which implies

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and the increment δx is no longer propagated in time. The 3D-FGAT method loses the dynamical aspect of the 4D-Var and is thus classified as a 3D variational method. The background state is still compared to observations at the observation time (i.e., the appropriated one) by Eq. (7). As the linear propagator is the identity matrix, the 3D-FGAT produces a static increment. According to Fisher and Andersson (2001), the valid time of the increment obtained by the classic 3D-FGAT (without the outer loop) is the central time of the window (not the initial time as in 4D-Var). Applying roughly the 3D-FGAT (or the 3D-VAR) analysis increment at the central time may cause a chock response of the model. To avoid this chock, the increment may be distributed around the central time for example by digital filters (Fillion et al. 1995), incremental digital filters (Gauthier and Thépaut 2001), or the incremental analysis update approach (Bloom et al. 1996).

To perform an outer loop with the 3D-FGAT, the increment has to be applied at the initial time, so to compute the misfit between the new first guess x_aⁿ⁻¹(t₀) and the observations at the observation time by Eq. (12). Using Eq. (13), we rewrite the cost function of Eq. (6), so the 3D-FGAT cost function at the iteration n is

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where δx_aⁿ = 0 for the first iteration and δx_aⁿ = δx_aⁿ⁻¹(t₀) − x_b afterward. The minimization of the cost function of Eq. (14) produces the analysis increment δx_aⁿ(t₀) and therefore an analysis x_aⁿ(t₀) = x_aⁿ⁻¹(t₀) + δx_aⁿ(t₀). This 3D-FGAT variant with the correction time placed at the beginning of the assimilation window will hereafter be simply referred to as 3D-FGAT. This variant makes the aforementioned filtering techniques that distribute the increment around the central time not appropriate as well. Moreover, as the data distribution is obviously more likely to be about the middle time, the expected consequence for this 3D-FGAT variant is that it will be penalized by the observations localized at the end of the window. Indeed, the time inconsistency between the propagated guess and the observations may introduce large errors in the innovation vector. This paper does not detail the differences between the classic 3D-FGAT and the 3D-FGAT with the increment applied at the beginning of the assimilation window. Nevertheless, their results should not be too different under conditions of weak enough impact from flow dependence over the assimilation window. Moreover, both would be deficient compared to 4D-Var under conditions of fast flow.

In the general nonlinear case, it is not possible to have an analytic expression of both 4D-Var and 3D-FGAT analyses as a function of the iteration of the outer loop. But, in the linear case, the 4D-Var solution is given by Eq. (8). The solution is found at the first iteration and does not change with the next iterations. However, this does not hold for 3D-FGAT (except if the model is the identity) as its solution depends on the iteration. The solution at the nth iteration is

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where 𝗞^FGAT is a gain matrix that does not contain the linear propagator 𝗠:

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Equation (15) shows that the convergence value of the iterative 3D-FGAT is not the 4D-Var solution even in the linear context. The first guess x_aⁿ⁻¹ is propagated by the model 𝗠, while the increment x_b − x_aⁿ⁻¹ is not propagated. It indeed shows there are two differences between 4D-Var (in terms of increments) gain of Eq. (9) and 3D-FGAT gain of Eq. (16). One is that background errors are flow dependent in 4D-Var since they are represented by 𝗠𝗕𝗠^T. The second one is that, in 4D-Var, the gain matrix included a backward propagation in time thanks to 𝗠^T. Neither of these flow-dependent effects occurs in 3D-FGAT. When the methods are compared, it is difficult to identify which of the two aspects predominates.

In the next section, we propose an example where only the second aspect occurs. The illustration focuses on the outer loop and shows that, even if the iterative 3D-FGAT analyses differ from the 4D-Var analyses, the use of the outer loop with 3D-FGAT is relevant. Actually, we illustrate that the interesting properties of the outer loop may be advantageous for the 3D-FGAT analysis. But, the outer loop can also cause a deficiency in the 3D-FGAT analysis in situations driven by rapid dynamics compared to the temporal size of the assimilation window.

a. Description of the numerical experiments

The numerical model described by Eq. (17) is used in the particular case of a constant velocity and for a circular domain of radius 6380 km, which corresponds to the earth’s great circle. The circle is divided into 445 evenly spaced grid points. In such a situation, Eq. (17) is reduced to a simple translation. The length of the time step is adjusted with the velocity so the field is translated over an integer number of grid points. This limits the numerical diffusion due to the numerical scheme and grants interesting properties of the propagator 𝗠 like 𝗠⁻¹ = 𝗠^T (see the appendix for further details).

The background x_b of the concentration of the passive tracer has a sinusoidal shape with a period equal to the half of the domain length, a mean equal to unity, and an amplitude of 0.1 (Fig. 1). A true state is built by adding to x_b a random Gaussian noise with a zero mean and a covariance matrix equal to the covariance matrix of the background errors. The standard deviation of the background errors is constant in space with its value set to 0.1. The correlations of the background errors are Gaussian with a length scale of 500 km. The knowledge of the true state allows us to build synthetic observations. The observation operator selects the value of the tracer’s concentration at the grid point where the observation is available. The observations are built by applying the observation operator to the true state and by adding a Gaussian distributed random noise with zero mean and a standard deviation also set to 0.1. There is no correlation applied between observation errors.

In our particular case of a constant velocity and a homogeneous covariance matrix of the background errors, according to Eq. (A3), the gain of Eq. (9) associated with 4D-Var becomes

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The relation 𝗕𝗠^T = 𝗠^T𝗕 can be derived from Eqs. (A2) and (A3) and leads to the equivalent expression of the following gain:

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According to Eq. (16), expressing the gain in the 3D-FGAT case, there is a direct link with this 4D-Var gain under the previously described assumptions:

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This relation shows that in our pure advection case with a homogeneous 𝗕 matrix, the analysis increment is the same for the 4D-Var and 3D-FGAT methods, but with a shift due to the adjoint (or inverse) model.

b. Using a nonlinear observation operator

A justification of using the outer loop with a 3D-FGAT assimilation scheme is the potential to relinearize each operator around a better estimate of the truth. In our case where the forecast model is linear, we illustrate the effect of the outer loop while using a nonlinear observation operator. The illustration simulates the assimilation of the radiance. According to the Beer–Lambert law, the radiation absorbed by the total atmospheric column of a chemical species with a concentration C(z) at the altitude z is proportional to the optical length τ :

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Here is the total column of the chemical species and σ is the absorption cross section of the species. As an example, choosing the ozone as the gas, τ represents the UV absorption.

Since there is no vertical component in our one-dimensional model, we simulate the total column by the state vector x. Assuming that τ₀ = 1 and σ = 1 leads to the nonlinear observation operator:

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In this experiment, we randomly choose 89 observations evenly spaced (every fifth grid point). The observations are assumed to be all measured at the end of a 3-h assimilation window (Fig. 2a). The observation operator consists here in (i) a projection of the state with the operator of Eq. (22) onto a space we will name observation space for simplicity, and (ii) a selection at the grid point where the observation is available. The velocity is constant with a value of 50 m s⁻¹.

We performed 10 iterations of the outer loop, refreshing the linearization of the observation operator at every iteration. For each iteration, we computed the standard deviation over the whole assimilation window of the current analysis minus the true state. This standard deviation is normalized by the one computed from the difference between the background and the true states. In this example, the standard deviation decreases with the iteration number and becomes almost constant after six iterations (Fig. 2c). The first analysis already provides a good estimation of the truth but with the increase in the number of outer iterations, the structures of the fields are better positioned and have a better amplitude compared to the truth (Fig. 2b). This illustrates the improvement coming from the update of the linearization origin in the outer loop when the assimilation problem is not linear.

c. The observation screening

As the 3D-FGAT requires the linearization of the assimilation operators, the increment has to be small enough to stay in the linearization validity region. A useful way to limit the amplitude of the increment is to proceed to a data selection by keeping those that are not too far from the background state (projected in the observation space). This is known as observation screening. The observation screening is also a mean to detect suspicious measurements and to remove the bad ones from the assimilated dataset.

The observations to be assimilated can be selected first of all when the background is integrated in time with the full propagation model and through the (possibly nonlinear) observation operators. The fact that removed data are far from the background means that some of them are doubtful or that the background is inappropriate. The outer loop produces a new first guess at each iteration. Expecting that the first guess is closer and closer to the truth, it can be used at each iteration to proceed to a new data selection. The removed observations that remain are therefore those that are doubtful, the first guess being no longer inappropriate. The outer loop here offers a chance to inject new observations in the assimilation process while still rejecting the bad ones.

We illustrate this property with the one-dimensional advection model. The variance of the background departure of Eq. (12) can be estimated as a sum of observation and background errors variances, assuming that the observation and the background errors are uncorrelated. In the observation screening, the square of the background departure may be considered as suspect when it exceeds its expected variance. In our example, the observation and background error standard deviations are both equal to 0.1: it makes a departure suspect if its standard deviation exceeds 0.14. To impose a looser criterion, we considered observations with a departure lower than 0.2.

Because of the construction of the twin experiment, most of the synthetic observations satisfy this criteria. We thus decided to keep the previous background state but to compute a new truth by adding to the previous one a random bias with a mean of −0.2 and a standard deviation of 0.1. The 89 computed observations are based on this new truth and are still evenly spaced and measured at the end of a 3-h assimilation window (Fig. 3a). This aims at simulating an atmospheric situation where the forecast model misses something, an important decrease in the concentrations over the whole domain in this particular example. In the proposed illustration, the observation operator exists only in a selection and the velocity is still constant with a value of 50 m s⁻¹.

We chose to compare two experiments: one using the same set of 69 observations selected thanks the background and one with a dynamic screening based on the current first guess of the iteration of the outer loop. As previously, we computed the normalized the standard deviation of the difference between the analysis and the truth. Even if the outer loop reduces this standard deviation when keeping the same number of observations, the reduction is increased when new observations are added to the assimilation process (Fig. 3b). In this example, the outer loop allows to add progressively 16 more observations over 89 available (Table 1), ignoring as expected the bad ones. For the first iteration of the outer loop, the analysis fields are by construction the same in the cases of a constant screening and an updated one. Most of the rejected observations have a value lower than the background value that is coherent with the negative bias used to build the observations (Figs. 3b,c). But only the case with the updated screening is able to catch these observations after several iterations of the outer loop. This illustrates the improvement coming from the update of the screening in the outer loop when the background state is inappropriate.

d. Situations governed by dynamical effects

In the pure advection case with a constant velocity, the model gets more and more different from the identity with the increase of the velocity. In the presence of high velocities, the 3D-FGAT assumptions reach the limits of their validity. We will show how the use of the outer loop acts in this case. To separate the dynamical effects from the outer-loop ones, we first begin with an assimilation example without the outer loop. Then the outer loop is added in a second example. Both examples use the previously described experiment framework. They are based on the assimilation of a single observation localized at the end of a 3-h assimilation window. The observation is realized at the longitude 164.2°.

1) Assimilation without the outer loop

As we only have a single observation, the 3D-FGAT increment distribution has a Gaussian shape due to the correlations of the background errors. The consequence of Eq. (20) is a different location for the maxima of the 3D-FGAT and 4D-Var increments. Within 4D-Var, the adjoint model is used to compute the position of the correction at the initial time in order for it to be transported at the proper location at the observation time. The correction of the 3D-FGAT is instead added at the observation location at the initial time and is then advected getting beyond the observation location at the observation time. This shows the risk of using the 3D-FGAT method when the dynamics is rapid compared to the size of the assimilation window. It is illustrated in Fig. 4 where the velocity of the forcing field is set to 200 m s⁻¹. This excessive value of the velocity is not relevant for atmospheric winds, but this value was chosen in order to have a strong advection during the assimilation window. In 3 h, a fluid particle at 200 m s⁻¹ covers 2160 km, which is more than 4 times the length scale of the correlation of the background errors. This means that the background field and the true state are shifted eastward by about 19.4° in 3 h.

To quantify the 3D-FGAT error, we compared it to the 4D-Var, computing the discrepancy between the analyses produced by the two methods, respectively x_a^FGAT and x_a^4D, by

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where ‖·‖ is the L₂ norm defined for a vector b of dimension n by

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The discrepancy is equal to 0% when the analyses are equal and is equal to 100% when the increments of the analyses are uncorrelated.

We evaluated the discrepancy as a function of the velocity. We let the velocity vary from 0.1 to 200 m s⁻¹. By modifying the velocity, we modify the model 𝗠. To measure how far is the model from the 3D-FGAT assumption, we built the Frobenius (or Hilbert–Schmidt) norm ‖·‖_F of 𝗠 − 𝗜, where 𝗜 is the identity matrix. The Frobenius norm of the 𝗔 matrix of dimension n is defined by

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where a_i,j is the element of 𝗔 on row i and column j. For the different values of the velocity, we computed the discrepancy between the 3D-FGAT and the 4D-Var analyses, as well as the norm = ‖𝗠 − 𝗜‖_F. We found that the discrepancy is a linear function of the velocity and of the norm for low values of the velocity (Fig. 5). At 10 m s⁻¹, a fluid particle covers 108 km in 3 h, which is just more than the size of the grid cell of about 90 km. This means that the concentration in a grid cell weakly influences the concentration in this cell after 3 h. The diagonal of 𝗠 is thus close to zero. This explains why the norm becomes almost constant when the velocity reaches 20 m s⁻¹. At the same time, the discrepancy still grows with the velocity. For velocities of 50 and 200 m s⁻¹, as we use in this paper, the discrepancy is respectively 66% and close to 100%.

2) Assimilation with the outer loop

The problem posed by the 3D-FGAT method when the tangent linear of the propagation model is far from the identity matrix is amplified when an outer loop is used. As illustrated above, the 3D-FGAT increment at the observation time may be situated away from the observation location. This means that 𝗛𝗠x_a does not differ much from 𝗛𝗠x_b. This case implies that the innovation vector of Eq. (12) for the second iteration of the outer loop is almost the same as in the first iteration. In the observation space, the effect of the first analysis is imperceptible. Assuming that 𝗛𝗠x_a = 𝗛𝗠x_b, Eq. (15) becomes

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where x_a¹ is the analysis obtained at the first iteration. As a consequence, each iteration will produce an increment of the same sign as the increment produced at the previous iteration. This increment is added to the previous one making the analysis differ more from the background (and from the first analysis) when the number of the iteration increases (Fig. 6a). It is possible to determine the convergence of the final increment (difference between the final analysis at the iteration n and the background). In the single observation example, the maximum of the total increment converges toward twice the value of the 4D-Var increment maximum for a velocity of 200 m s⁻¹ (Table 2).

When the dynamics is less rapid, the propagation model is closer to the identity matrix but the behavior remains similar. When the discrepancy is lower than 100%, the 3D-FGAT increment propagated at the observation time is not null at the observation grid point. This makes the innovation vector change from one outer loop to another. Nevertheless, each iteration of the outer loop adds an artificial supplementary increment to the previous ones, as illustrated by Fig. 6b for a velocity of 50 m s⁻¹. The difference with high velocity values is the fact that the convergence of the total increment maximum is obtained for a lower value (Table 2).

a. The assimilation suite

The assimilation system used in this study is based on the European Centre for Research and Advanced Training in Scientific Computation (CERFACS) Valentina assimilation framework and the Météo-France comprehensive three-dimensional CTM Mocage. Valentina is an extension of the Mocage–Palm system (Massart et al. 2005) developed jointly by CERFACS and Météo-France in the framework of the FP5 European project Assimilation of Envisat Data (ASSET; Lahoz et al. 2007). It is based on the CERFACS Palm software (Buis et al. 2006) that allows it to be coupled with different CTMs. The flexibility offered by Palm makes Valentina a system that offers several assimilation options, like the choice of the variational method (3D-FGAT or 4D-Var), the choice of the representation for covariances of the background errors and the choice of the analyzed domain (regional or global).

The CTM Mocage covers the planetary boundary layer, the free troposphere, and the stratosphere. It provides a number of optional configurations with varying domain geometries and resolutions, as well as chemical and physical parameterization packages. Mocage is currently used for several applications, with recent examples in chemical weather forecasting (Dufour et al. 2004), chemistry–climate interactions (Teyssèdre et al. 2007), and intercontinental transport of ozone and of its precursors (Bousserez et al. 2007).

The first version of Valentina using Mocage, as it was originally implemented for the ASSET project, provided good quality ozone fields as compared with independent measurements with errors of the same order as those produced by several other assimilation systems (Geer et al. 2006). To improve the Valentina assimilation suite, several changes have been recently made in the characterization of the forecast errors (Massart et al. 2007; Pannekoucke and Massart 2008). To increase the consistency of the analysis, several diagnostics were added (Massart et al. 2009) as discussed by Desroziers et al. (2005). Recently, the outer loop has been implemented for the incremental methods.

b. Description of the experiment

In this study, we worked with a 2° by 2° global version of Mocage, with 60 vertical levels (from the surface up to 0.1 hPa). The meteorological forcing fields are provided by the operational ECMWF numerical weather prediction model. To compute the ozone fields, we adopted the linear ozone parameterization developed by Cariolle and Teyssèdre (2007) in its latest version. This parameterization is based on the linearization of ozone production–destruction rates using an altitude–latitude chemical model.

The forecast error covariance matrix of the Valentina assimilation suite is split into a correlation matrix and a diagonal matrix filled with the forecast error variances (square of the forecast error standard deviations). The correlation matrix is divided into a horizontal operator and a vertical operator. These operators are both modeled using a diffusion equation, which is a practical way to generate correlation shapes and in particular Gaussian shapes (Weaver and Courtier 2001). For this study, we generated only Gaussian correlations that can be characterized by their length scale. Horizontal correlations of the forecast errors are computed using a constant length scale of 220 km corresponding to a distance of 2° at the equator. Vertical correlations are computed using a constant length scale of 0.35 (in units of the logarithm of the pressure).

The assimilated data come from the Microwave Limb Sounder (MLS) instrument. This instrument has been flying onboard the Aura satellite in a sun-synchronous polar orbit since August 2004. Vertical profiles of several atmospheric parameters are retrieved from the millimeter and submillimeter thermal emission measured at the atmospheric limb (Waters et al. 2006). Measurements are performed between 82°S and 82°N, with a long-track resolution that varies from about 165 km to nearly 300 km. For our study we have used the latest version (v2.2) of the MLS ozone product (see Massart et al. 2009 for more details).

The experiment we carried out was a reanalysis of the global atmospheric ozone performed for the fall of 2007. The assimilation scheme used for this was the 3D-FGAT method with the increment applied at the beginning of 3-h windows and three outer-loop iterations.

c. Assimilation results

We present hereafter only the reanalysis results obtained for 27 August 2007 and for the assimilation window that spans from 0300 to 0600 UTC, when a particular behavior of the assimilation was encountered. We focus on the region 60°–15°S, 100°–10°W at the 10-hPa pressure level (which corresponds to an altitude of approximately 30 km). This region of interest is chosen because there is an important gradient in the wind strength for the selected date, with values ranging from 0.6 to 107 m s⁻¹ (Fig. 7). Moreover, during the assimilation window, the satellite flies across this region twice, once at the beginning of the window between 0300 and 0307 UTC, and then at the middle of the window between 0435 and 0446 UTC (Fig. 8).

As expected, the 3D-FGAT increments of the first outer-loop iteration are located around the observation location (Fig. 8a). Their spread is due to the correlation of the background errors. As described previously, these increments are added at the initial time of the assimilation window (0300 UTC) to the background state, in order to compute the first analysis. But, with a wind of about 100 m s⁻¹ at 60°S, the increment is propagated eastward at approximately 540 km at 1.5 h later, when the second set of data is measured. At 60°S, the distance of 540 km represents a shift of about 10° in longitude. At 45°S, it represents a shift of about 7°. This means that at these latitudes, between 0435 and 0446 UTC, the background and the first analysis are similar at the observation location of the second set of observations. The innovation vector for the second iteration of the outer loop is computed using the first analysis as a first guess, as shown by Eq. (12). As a consequence, for the second set of observations (measured between 0435 and 0446 UTC), in the latitudes between 45° and 60°S, the innovation vector of the second iteration is the same as in the first iteration. This situation is similar to what we saw with the example on the circle when the value of the velocity was 200 m s⁻¹ (Fig. 6a). The consequence is that the second iteration of the outer loop computes an increment in the region where the strength of the wind is important of the same order as at the first iteration (Fig. 8b). Where the strength of the wind is lesser, the increments brought by the second outer loop are negligible.

Concerning the first set of observations, as it is measured at the beginning of the assimilation window, the propagation of the first analysis at the time of these observations is close to the identity matrix. The norm ‖𝗠 − 𝗜‖_F is low and the 3D-FGAT solution is close to the 4D-Var one. Thus, the analysis computed for the second iteration of the outer loop produces an increment with very small values.

The behavior of the third iteration of the outer loop is similar to the second one. At the latitudes where the strength of the wind is important compared to the size of the assimilation window, the outer loop amplifies the increment. The consequence is an overestimation of the total correction applied to the ozone concentrations in these regions (Fig. 8c). Moreover, when the analyzed ozone field is compared to the observations (at the observation time), the assimilation process seems not to bring any improvement at these latitudes.