# Coupled Stratospheric Chemistry–Meteorology Data Assimilation. Part II: Weak and Strong Coupling

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

_{3}, NO

_{2}, and SO

_{2}) using the coupled tropospheric model WRF-CHEM with an ensemble based approach [38]. Coupling can also occur through coupled observation operators. For example, infrared channels of operational meteorological satellites are sensitive to ozone and CO

_{2}and can benefit from using an ozone assimilation [39] and a CO

_{2}assimilation [40].

^{−1}) was found on zonal wind with no reduction of error standard deviation [47]. These rather unsuccessful results conducted in an operational context suggested that additional studies were necessary. Using an ensemble Kalman filter and an intermediate-complexity model, Milewski and Bourqui [48] demonstrated that information about the ozone-wind cross-covariance is essential in constraining dynamical fields when ozone only is assimilated. Moreover they showed that a further reduction in error can be obtained with an ensemble Kalman smoother [49]. In a series of studies using 4D-Var and the ensemble Kalman filter, Allen et al. [50,51,52] showed that poorly-specified observation error could lead to an increase in RMS wind error, that observational coverage is important such that wind extraction could be improved if several chemical tracers were used, and that the balance between wind and temperature could be offset by the wind recovery from tracer measurements. We should note that the wind extraction from tracer observation is part of a more general class of joint state-parameter estimation problems (e.g., [53,54]).

_{3}, CH

_{4}, and N

_{2}O to correct the winds.

## 2. Extension of 3D-Var and 4D-Var for Chemical—Meteorological Coupling

- 3D-Var: In this case, all observations collected over the six-hour assimilation window are assumed to be valid at the central time. Observation departures from the model state (called innovations) are computed with respect to the background state valid at the central time of the window [56].
- 3D-FGAT (First Guess at Analysis Time): This scheme is a variant of 3D-Var in which the innovations are evaluated by comparing each observation with the model output valid at the observation time [57] (actually closest to a 1-hour bin). This is the default configuration used in all 3D-Var experiments in this study.

_{3}, CH

_{4}, NO

_{2}, N

_{2}O, HNO

_{3}, and ClONO

_{2}). As far as unobserved constituents that have correlated errors with observed constituents, their minimum variance estimate can be obtained off-line after the minimization.

#### 2.1. Analysis Splitting between Observed and Unobserved Variables

_{3}, N

_{2}O, NO

_{2}, CH

_{4}, and HNO

_{3}. We will derive in this section a computational simplification that allows splitting the analysis into observed and unobserved variables parts.

**Z**be the complete chemical-meteorological state vector be decomposed into observed variables

**X**and unobserved variables

**U**, i.e.,

**y**denotes the observation vector (i.e., all observations of all observed variables at a given time), H is the observation operator,

**R**the observation error covariance matrix, and

**B**is the full state background error covariance matrix that can be decomposed into:

**Z**we found that the minimization of $J(Z)$ can be split into two parts: A minimization of the cost function involving only the observed variables and observations, which takes the form:

**U**in either strongly-coupled or weakly-coupled data assimilation system. The analysis increment in a strongly-coupled data assimilation system would use ${U}^{a}$ (Equation (5)) as part of the initial condition ${Z}^{\mathrm{initial}}={({X}^{a},{U}^{a})}^{T}$ for a coupled model. In a weakly-coupled data assimilation system, we would use ${U}^{f}$ (instead of ${U}^{a}$) to initialize the unobserved space, and furthermore ${X}^{a}$ would be obtained from an uncoupled analysis. That means that, for example, in weakly-coupled chemistry–meteorology data assimilation, ${X}^{a}={({\mathsf{\mu}}^{a},{\mathsf{\chi}}^{a})}^{T}$, where $\mathsf{\mu}$ are meteorological and $\mathsf{\chi}$ chemical variables so that in the coupled model the initial condition is given by ${Z}^{\mathrm{initial}}={({\mathsf{\mu}}^{a},{\mathsf{\chi}}^{a},{U}^{f})}^{T}$.

#### 2.2. General Description of the 3D-Var-CHEM

**X**= (ψ, χ, T, ln(q), c

_{1}, …, c

_{N}, p

_{s})

^{T}, where $\mu $ is the streamfunction, $\chi $ the velocity potential (not the confuse with $\mathsf{\chi}$ used in the previous sub-section), T the temperature, q the (tropospheric) water vapor mixing ratio, p

_{s}the surface pressure and N observed chemical constituent with mixing ratios c

_{1}, …, c

_{N}. The state vector in 3D Var-Chem is such that all 3D fields are grouped together, followed by the 2D field p

_{s}. As explained in Section 2.1, the state augmentation is limited only to observed variables/species.

**L**times a vector

**X**, can be obtained as a sequence of operators, without the need to store any large matrices. This property arises principally from the assumption that the horizontal error correlation is assumed to be homogeneous and isotropic on the sphere. For such correlations, the spectral representation is diagonal in spectral space (see for example [56,62,66,67]). The sequence of operations then becomes as follows: (1) We multiply the spectral representation of the state $X$ with the square root of the spectral coefficient of the correlation model ${\mathsf{\Lambda}}^{1/2}$ (which is a diagonal matrix for separable models and bloack diagonal for non-separable correlation model); (2) perform a transform from spectral to physical space. First using a spectral transform $S$ which gives a field on the Gaussian grid, and then followed by an interpolation $G$ from the Gaussian grid to a regular latitude–longitude grid; (3) multiply the resulting fields by the error variances (that takes the form of a diagonal matrix of error standard deviation $\mathsf{\Sigma}$); and (4) using balance operators

**M**(to be define below), transform the primary fields into fields of physical significance accounting for cross-correlations between them. This is how we obtain, for example, the velocity potential from the stream function and an unbalanced velocity potential. This last operation is obtained through a balance operator. In the end, the square root of ${B}_{XX}$ is given as $L=M\mathsf{\Sigma}GS{\mathsf{\Lambda}}^{1/2}$, a sequence of operators.

#### 2.3. Balance Operators

_{2}O, CH

_{4}) or chemically-related species such as (O

_{3}, NO

_{2}), but we have not done so here.

**C**, is actually represented spectrally as $C=S\mathsf{\Lambda}{S}^{-1}$ where

**S**and ${S}^{-1}$ are the spectral direct and inverse transform and $\mathsf{\Lambda}$ is a diagonal or block-diagonal ($nlev\times nlev$) matrices of spectral coefficients. For computational efficiency, the balance operators in

**M**are simplified as block diagonal matrices ($nlev\times nlev$) for each latitude with an error variance that depends on height and latitude (using a Legendre polynomial expansion).

**M**, which turns out to be easy to obtain as:

#### 2.4. D-Var Tracer Extension

## 3. Error Statistics

#### 3.1. Estimation of Error Variances by Autocorrelation of Innovations along the Satellite Track

^{−15}and is distinctively different from the extrapolated intercept of the spatially correlated part, estimated to be around 8 × 10

^{−15}. Such a separation of values at zero distance is observed at all levels and for all species. This supports our assumption that the observation error is either spatially uncorrelated or that the spatial correlation length is much shorter than the background error correlation.

_{4}, obtained from HL method are displayed in the left panel of Figure 3. We note that the MIPAS CH

_{4}observation error variances are significantly larger than those of the (model) background errors except in the region between 2 and 0.5 hPa. This indicates that MIPAS CH

_{4}observations will have a small impact on the analysis, the main impact region is limited to 2–0.5 hPa and also in the lower stratosphere between 100 and 50 hPa.

_{3}, CH

_{4}, N

_{2}O, NO

_{2}, HNO

_{3}, and H

_{2}O that were assimilated in the course of this study. We note for instance that for O

_{3}the gain is about 0.2 in the lower stratosphere and steadily increases to about 0.6 in the upper stratosphere. A similar situation was found for the long-lived species CH

_{4}and N

_{2}O. However, the NO

_{2}gain is close to one in the upper stratosphere, indicating that the model has a small impact at these altitudes. As for HNO

_{3}, the gain increases with height and reaches a maximum value of 0.8 at 4 hPa, then decreases with altitude. Chemical water vapor (H

_{2}O) is presented in terms of the log of concentration.

#### 3.2. The Canadian Quick Covariance Method

_{4}and H

_{2}O above 70 hPa and NO

_{2}above 10 hPa are (roughly) separable as well as N

_{2}O in the entire stratosphere (although the correlation field is very noisy). However, the correlations for O

_{3}and HNO

_{3}at large scales (for wavenumbers smaller than 20) are not (horizontally/vertically) separable. These results suggest that a chemical specie has a non-separable correlation model if its chemical time-scale lies within the statistical sampling time-range which here is anything larger than 6 h and smaller than one month, otherwise the correlation of the species is separable.

_{2}and HNO

_{3}mixing ratio increments [88]. Figure 4 presents the mean analysis increment for HNO

_{3}.

_{3}with the new statistics is larger and self-organized, indicating vertical descent of HNO

_{3}in the polar vortex [88], while the old statistics give rather random results with numerous small-scale features. The analysis increments for chemically active species such as O

_{3}and NO

_{2}(Figure S6 in Supplementary Material) also appear to be larger and physically coherent, while those of passive tracer (CH

_{4}, N

_{2}O) are not changed significantly, remaining spatially random with both old and new statistics, with the difference that the increments with the new statistics are of somewhat larger scales.

#### 3.3. Cross-Covariance Estimates

_{3}as a tracer and the streamfunction relates to the tracer-wind coupling discussed in Section 2.3 Part I, which has been an elusive coupling relationship to obtain diagnostically [89,90,91] (see also discussion in Section 7). It has been argued that in regions of Rossby wave breaking, potential vorticity is correlated with ozone in the lower stratosphere where O

_{3}behaves as a chemical tracer. Figure S7 (Supplementary Material) shows scatter plots of O

_{3}concentration with streamfunction values between 10 and 100 hPa for March 2003 for different latitude bands. We note, however, that streamfunction and ozone have no significant correlation except at the highest latitudes in the Northern Hemisphere. We, thus, make the simplification and approximation that globally, the correlation between O

_{3}and streamfunction can be neglected. This also implies that the balance operator G (Equation (8)) can be neglected. Regarding Equation (14) we, thus, make the approximation that:

## 4. Harmonization of AMSU-A Radiances with MIPAS Temperatures

## 5. The Added Value of the Assimilation of Limb Sounding (MIPAS) Temperatures

_{4}above 3 hPa. Thus, we see that the presence of AMSU-A temperatures in the assimilation actually degrades the vertical structure, because of the coarse vertical resolution sensitivity of the associated channels, which is apparent in the transport of chemical species in regions of strong vertical concentrations.

## 6. Weak Coupling Assimilation due to Ozone–Radiation Interaction

**A**between ozone and temperature. The associated background (or model) error covariance is given by Equation (19) and using the operator

**A**. We have shown already in Figure 5 that ozone–radiation interaction increases the correlation between temperature and ozone between 10 and 100 hPa (in the northern latitude summer). Consequently, the error cross-covariance and its effect on the error variance of ozone is increased, and this is what it is observed in the right panel of Figure 12.

## 7. Strongly Coupled Temperature–Ozone Assimilation with 3D-Var

_{3}and AMSU-A temperatures (all channels) for a period of two weeks, from 17 August to 4 September 2003. Figure 17 shows the verification over the globe in the three case: univariate (blue), multivariate with the balance operator ${A}^{CQC-NMC}$ (red), and multivariate with the LINOZ balance operator ${A}^{LINOZ}$ (grey dots).

## 8. Strongly Coupled Tracer–Meteorology Assimilation with 4D-Var

**X**are solutions of:

**V**is the horizontal velocity vector at coordinate x and time t.

**X**is the Lagrangian solution of the flow, and since a tracer is a Lagrangian-conserved quantity, the concentration of a chemical tracer c depends only on

**X**, i.e., c = c(

**X**). On the other hand, a non-divergent flow can be described entirely by a streamfunction ψ:

**V**= (u,v), such that ∇ ⋅

**V**= 0. However, since streamfunctions also have the property that $V\cdot \nabla \psi =0$, in a steady-state case where ${\partial}_{t}\psi =0$, the material derivative of ψ is zero. In this case, the streamfunction ψ is constant following the material particles, and, thus, the streamline and streamfunction coincides, and one could, thus, use the streamfunction as a proxy for the concentrations.

**X**. From a statistical point of view, the cross-covariance 〈uc〉, 〈vc〉 between wind and the concentration plays a fundamental role in our ability to infer information about wind from concentration. If these cross-covariances are zero, statistical inference is not possible. Thus, we can see that statistical inference of winds from tracer in a steady-state non-divergent flow depends on gradients of concentration error variance.

_{3}, CH

_{4}, N

_{2}O, and all three together. The results shown at 10 hPa in Figure 19 indicate the additive nature of the wind increments as the three species lead to different impacts at different locations. Analysis wind increments obtained at 50 and 100 hPa are displayed in Figures S16 and S17 (Supplementary Material). The differences in the increments can be explained “mechanistically” by differences in the distribution of the constituents at different levels. Figure S18 (Supplementary Material) shows that the distribution of N

_{2}O is more homogenized than that of O

_{3}at 100 hPa. Ozone, generated in the tropical lower stratosphere, is transported in the Southern Hemisphere on a relatively short time scale. Gradients in the ozone field are more important than the gradient of N

_{2}O, and, thus, provide more information about the underlying winds. When observations are present, the presence of these gradients yields the most significant wind increments. Nitrous oxide observations (N

_{2}O) are also involved, but the weaker wind gradients in this field make it more difficult to accurately locate the displacement, which contains the wind information.

^{−1}to 3 ms

^{−1}agrees, for the most part, with the zonal wind correction in Figure 20, except near 20 hPa where the difference between 4D-Var experiments is about 3 ms

^{−1}while the difference with radiosonde observations indicate a correction of about 1 ms

^{−1}. We also note large mid-latitude corrections especially in the Southern Hemisphere just outside the polar vortex, in the surf-zone, i.e., the region of Rossby wave breaking.

_{3}, CH

_{4}, N

_{2}O) is a second-order effect and is more difficult to assess, as the 4D-Var assimilated those constituents to produce a wind correction. For ozone, differences appear mainly in the winter hemisphere (Southern Hemisphere) where dynamical processes are more important. Figure S19 (Supplementary Material) shows the comparison of ozone from the assimilation against MIPAS O

_{3}observations for the period 20 September to 5 October. Below 10 hPa in the Southern Hemisphere mid-latitudes and polar regions, the results of 4D-Var shows a slight degradation at about 100 hPa. In the case of methane and nitrous oxide, differences between both analyses appear in the tropics and are mainly driven by changes in the zonal wind (not shown).

## 9. Summary and Conclusions

_{3}, and HNO

_{3}on large scales (wavenumber 20 and smaller). However, the resulting horizontal correlation lengths appear to be too small. With the CQC approach, we also computed cross-covariances between ozone and temperature, and showed that it contains signals not only from photodissociation and ozone–radiation interaction but also transport, which is undesirable. The cross-covariance should in fact be computed between ozone and the residual temperature unexplained by the mass field (i.e., unbalanced temperature) rather than the temperature itself, but this would requires additional development of the CQC method.

_{3}, CH

_{4}and N

_{2}O) observations from the limb sounder MIPAS to infer winds in the stratosphere. Inference on winds can be mechanistic in nature, that means recovering wind information from a time series of concentrations (e.g., a uniform concentration has no mechanistic capability in inferring winds). The inference can also be statistical in nature, where gradients in concentration error variances introduces cross-covariances between winds and chemical tracers [43,44,48]. Our experiments demonstrated the importance of having correct chemical background and observation error covariances, thus, supporting the statistical inference nature of the problem. The use of multiple tracers was also shown to be complementary, as the horizontal distribution of concentration gradients and vertical distribution of background error is different for different chemical tracers. Overall, an improvement in the tropical zonal winds was found in the lower stratosphere and a large portion of the troposphere, as assessed with radiosonde observations. A zonal-wind increment of about 2.5 ms

^{−1}was also found in the surf-zone above 5 hPa but it is unclear if this helped the transport of chemical constituents, possibly due to the fact that assimilating chemical tracers result in a second-order effect, which is not easily detectable. We also observed the buildup of a temperature bias in the tropical lower stratosphere (at 20 hPa) associated with the tropical wind correction—a wind correction that is supported by the radiosonde observations.

## Supplementary Materials

**Figure S1**. Flow chart covering the main steps and options of the 3D-Var-Chem.

**Figure S2**. Scalar gain for O

_{3}, CH

_{4}, N

_{2}O, HNO

_{3}, NO

_{2}, and ln(H

_{2}0).

**Figure S3**. Background vertical error correlation power spectra from 6h-difference method.

**Figure S4**. Background error variance from 6hr-difference method.

**Figure S5**. Horizontal correlation length.

**Figure S6**. Mean analysis increment for O

_{3}, CH

_{4}, N

_{2}O, NO

_{2}.

**Figure S7**. Scatter of O3 and streamfunction values between 10 and 100 hPa for the month of March 2003.

**Figure S8**. Cross-correlation between ozone and temperature derived from 24-h difference method for July 2003.

**Figure S9**. Horizontal coverage of AMSU-A profiles in six hours.

**Figure S10**. Sensitivity matrix of brightness temperature over temperature for channels 10–14 of AMSU-A.

**Figure S11**. Mean analysis increment at 10 hPa for the month of September 2003.

**Figure S12**. Zonal mean analysis increment for September 2003.

**Figure S13**. Global verification of observation-minus-forecast temperatures for different forecast lead time.

**Figure S14**. Coefficient of the LINOZ scheme for September.

**Figure S15**. LINOZ climatology for September.

**Figure S16**. Same as Figure 19 but at 50 hPa.

**Figure S17**. Same as Figure S19 but at 100 hPa.

**Figure S18**. Analysis of N

_{2}O and O

_{3}at 100 hPa on 11 August 2003, 00 UTC.

**Figure S19**. OmP ozone comparison against MIPAS for the 3D-Var assimilation cycle and 4D-Var for the period of 20 September to 5 October 2003 over the South Pole region and Southern Hemisphere mid-latitudes.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

3D-Var | Three-dimensional variational analysis |

3D-Var-Chem | 3D-Var coupled meteorology–chemistry |

4D-Var | Four-dimensional variational analysis |

AMSU | Advanced Microwave Sounding Unit |

BASCOE | Belgian Assimilation System for Chemical ObsErvations |

CMC | Canadian Meteorological Center |

CQC | Canadian Quick Covariance method |

DU | Dobson Unit |

ECCC | Environment and Climate Change Canada |

ECMWF | European Center for Medium Range Forecasting |

EOS | Earth Observing System |

FGAT | First guess at appropriate time |

GEM | Global Environmental Multiscale |

GEM-BACH | GEM Belgian Atmospheric CHemistry |

HALOE | HALogen Occultation Experiment |

HL | Hollingsworth–Lönnberg method |

IR | Infrared |

LINOZ | LINearized model for Ozone |

MDPI | Multidisciplinary Digital Publishing Institute |

MIPAS | Michelson Interferometer for Passive Atmospheric Sounding |

NASA | National Aeronautics and Space Administration |

NH | Northern Hemisphere |

NMC | National Meteorological Center method |

NOAA | National Oceanic and Atmospheric Administration |

NWP | Numerical Weather Prediction |

O-P | Observation minus Prediction (or forecast) |

RMS | Root Mean Square |

SH | Southern Hemisphere |

TOMS | Total Ozone Mapping Spectrometer |

UARS | Upper Atmosphere Research Satellite |

WRF-CHEM | Weather and Research Forecasting model coupled with Chemistry |

## Appendix A. Derivation of Analysis Splitting between Observed and Unobserved Variables

_{b}term to a simple inner product term. The way to accomplish this transformation of variable is by factoring

**B**into square root and invertible matrix

**S**:

_{b}term then simplifies to:

**B**covariance matrix leads to a decomposition of the form:

**B**is then of the form:

**S**, let it first be represented in the form:

**S**. So letting $e=0$ in Equation (A8) we first get:

**L**, thus, we have:

**g**is the square-root of the inverse of

**G**:

## Appendix B. Geometric Interpretation of the Derivation of the Balance Equations

**x**and

**y**are random vectors. The effect of an inner product in a Hilbert space of random variables is, thus, to create a non-random variable. In Equation (A25), $\langle x,y\rangle $ is a matrix where each entry is non-random. The square of the norm is then the variance, ${\Vert x\Vert}^{2}=\mathrm{var}(x)$, and the correlation matrix $\mathsf{\theta}$, between variables

**x**and

**y**is obtained as $\mathrm{cos}(\mathsf{\theta})=\langle x,y\rangle /\left(\Vert x\Vert \Vert y\Vert \right)$. Therefore, the uncorrelated random variables $\mathrm{cov}(x,y)=0$ are orthogonal, i.e., $\langle x,y\rangle =0$.

## Appendix C. Error Covariance from the LINOZ Scheme

^{−6}.

^{16}molecules cm

^{−2}. The overhead number of molecules of ozone is:

^{−3}. The volume mixing ratio is the ratio of the number density of the gas over the number density of (dry) air, i.e.,

^{−1}), ${N}_{a}$ is Avogadro’s number (equal to 6.02252 $\times $ 10

^{23}molecules mol

^{−1}), and ${\rho}_{A}$ is the density of air, the overhead number of ozone molecules can then be rewritten as:

**Figure A2.**Point-by-point, $(\lambda ,p)$, scatter of ${c}_{3}(p)/{c}_{2}(p)$ with ${\overline{\mathrm{O}}}_{3}$.

#### Extension with the Photochemical Term ${c}_{4}$

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**Figure 1.**Observation component of the cost function for ozone assimilation as a function of iteration. Solid line is associated with the value of J

_{o}of the first inner loop and the dashed line the value of J

_{o}of the second inner loop.

**Figure 2.**Spatial autocovariance of innovation for MIPAS CH

_{4}at 63 hPa. The abscissa shows the lags between orbits (each orbits are separated by about 530 km along the satellite track). The red squares represent the sample autocovariance values, and the dashed curve is a piecewise linear interpolation between the sample points. Note that the sample covariance at zero distance separation is at the top of the graph (near 36 × 10

^{−13}), and the dashed line is extrapolated at lag 0.

**Figure 3.**Estimated error variance for CH

_{4}/MIPAS as a function of height (pressure in hPa). (

**Left panel**) The estimated background error variance (blue with circles) and observation error variance (red with squares) using the HL method. (

**Right panel**) Three different estimates of observation error variance. The blue curve with squares is the estimate given by the instrument team (i.e., the instrument error), the red curve with squares is the observation error variance obtained from the HL method (note it is identical to the red curve in the left panel), and the green curve with squares is the observation error variance estimated from the Desroziers method [81] (single iterate).

**Figure 4.**Zonal mean analysis increment for HNO

_{3}as function of height (in hPa). (

**Left panel**) Using the first guess or old statistics. (

**Right panel**) Using the new statistics consisting of CQC correlation and HL error variances. The value of the increment should be scaled by 10

^{−9}volume mixing ratio. Note that the contour interval on the left panel are much smaller than on the right panel, and only one shaded blue contour (values between −0.5 and −1.0) appears in the left panel.

**Figure 5.**Cross-correlation between ozone and temperature derived from six-hour differences (i.e., CQC method) for July 2003. (

**Left panel**) The non-interactive ozone-radiation run of GEM-BACH and (

**right panel**) for an interactive ozone-radiation run.

**Figure 6.**Balance operator between ozone and temperature for July 2003. (

**Left panel**) ${A}^{CQC-NMC}$, which uses CQC for correlations and the NMC method for temperature error variance, and (

**right panel**) ${A}^{LINOZ}$, as derived from the LINOZ scheme (derivation in Appendix C).

**Figure 7.**Mean (lower curves) and standard deviation (upper curves) of the AMSU-A radiance observations minus the forecast (6 h) for channels 11 to 14 (

**upper panel to lower panel**). In blue are the results using the standard CMC bias correction scheme, which uses only the model in the stratosphere, and in red using only MIPAS temperature in the stratosphere.

**Figure 8.**Global verification (observation-minus-forecast) of temperatures as function of height (in hPa) for two assimilation runs. In blue is the assimilation of AMSU-A only, and in red the assimilation of MIPAS temperature and AMSU-A. All AMSU-A data are processed with the new bias correction. The left panel illustrates verification against MIPAS temperatures, and (

**right panel**) verification against HALOE temperatures. The squares on the far right of the panels indicate a significant Student’s t-test with 95% (or higher) confidence interval, and the dots on the far right of the panels indicate a significant the Fisher test of variances with a 95% (or higher) confidence interval. These markers are red (blue) if the red (blue) experiment shows an improvement.

**Figure 9.**Global verification (observation-minus-forecast) against HALOE temperatures for two assimilation runs. In blue is the assimilation of AMSU-A, and in red is the assimilation of MIPAS temperatures only. Tests of significant differences are the same as in Figure 8.

**Figure 10.**Global verification (observation-minus-forecast) of temperature as for two assimilation runs. In red, is the assimilation of MIPAS temperature and AMSU-A but no stratospheric channels, and in blue is the assimilation of MIPAS temperatures with all the AMSU-A channels. (

**Left panel**) The verification against MIPAS temperatures, and (

**right panel**) the verification against HALOE temperatures. Tests of significant differences are the same as in Figure 8.

**Figure 11.**Same as Figure 10, but for verification of ozone MIPAS on the left and ozone HALOE on the right.

**Figure 12.**Global verification of the impact of ozone radiation interaction. All runs with the assimilation of MIPAS temperatures and AMSU-A channels 1–8 only. (

**Left panel**) The temperature impact, (

**right panel**) and the ozone impact. Runs with no ozone–radiation interaction are in blue, and with ozone–radiation interaction in red. Test of significant differences of statistics are the same as in Figure 8.

**Figure 13.**Fifteen-day forecast of temperatures at 70 hPa verified against analyses over the South Pole region resulting from the assimilation of MIPAS temperature and ozone. The curves in blue are from using the BASCOE chemistry, in red using the LINOZ linearized ozone chemistry, and in green using a climatological ozone. Note that, here, the bias is depicted with dash lines, while the solid lines represent the root mean square (RMS) error (not the standard deviation).

**Figure 14.**Total column ozone (DU) (analysis) over the South Pole region on 3 October 2003 resulting from the assimilation of MIPAS temperature and ozone. (

**Left panel**) Experiment using the BASCOE chemistry scheme. (

**Right panel**) Experiment using the LINOZ linearized ozone chemistry scheme.

**Figure 15.**Time series of ozone at 70 hPa over the South Pole region resulting from the assimilation of MIPAS temperature and ozone. The blue curve is from using the BASCOE chemistry scheme, and the red curve using the LINOZ linearized ozone chemistry scheme.

**Figure 16.**Anomaly correlations at 10 (red), 50 (green) and 100 (purple) hPa in the southern hemisphere (20–90S) for ozone-radiation interactive (dashed lines) and non-interactive ozone (solid lines) experiments.

**Figure 17.**Multivariate temperature-ozone assimilation. Univariate ozone and temperature assimilation (blue), multivariate assimilation performed with the CQC-NMC balance ${A}^{CQC-NMC}$ (red) and with LINOZ balance ${A}^{LINOZ}$ (grey dots). The solid lines denote average differences (biases) and the dashed lines indicate the standard deviations (by ±σ), except for ${A}^{LINOZ}$ where we use only grey dots for both metrics. (

**Left panel**) The temperature O-P (observation minus six-hour forecast) statistics from comparisons to MIPAS observations. (

**Right panel**) The ozone O-P statistics. Note that to simplify we have not presented the significant tests of differences of statistics because it would have been cumbersome to illustrate with three pairs of experiments.

**Figure 18.**Wind analysis increments in response to MIPAS CH

_{4}observations obtained with (

**a**) the first estimate of background-error statistics for chemical constituents, and (

**b**) the new statistics estimated using the Hollingsworth–Lönnberg method. Contours are the wind amplitude in m/s and arrows indicate the wind direction. Results are shown here at the 100-hPa level.

**Figure 19.**Wind analysis increments at 10 hPa obtained by assimilating CH

_{4}(

**top left**), O

_{3}(

**top right**), N

_{2}O (

**bottom left**), and all three species (

**bottom right**). Contours are the wind amplitude in m/s and arrows indicate the wind direction.

**Figure 20.**Impact of assimilating stratospheric chemical tracers on tropospheric tropical zonal winds. Verification against radiosonde observations over the tropical region (20° S–20° N) of observation minus 6-h forecast for the period 15 August to 5 October 2003. The results in red correspond to a 4D-Var assimilation experiment with assimilation of ozone, methane, and nitrous oxide. Results in blue are 4D-Var experiments but without assimilation of the long-lived species. The squares on the far right of the figure indicate a significant Student t-test of means with 95% confidence interval, and the dots indicate a significant the Fisher test of variances with a 95% confidence interval. These markers are red, indicating an improvement.

**Figure 21.**Difference between the (analysis) wind magnitude (in m/s) obtained from two 4D-Var assimilation cycles executed with and without the assimilation of ozone, methane and nitrous oxide. The results are averaged over the period 15 August to 5 October 2003. The zonal mean average is shown here.

**Figure 22.**OmP temperature time series between the radiosondes and the 3D-Var (blue) and 4D-var (red) assimilation cycles at 20 hPa in the North Hemisphere.

Variable Type | Statistical Parameters | Statistical Assumption and Methods | |
---|---|---|---|

Observation Error | Background Error | ||

meteorological | variances | innovation-based | combination of innovation-based and lagged-forecast (NMC) methods |

correlations | spatially uncorrelated | lagged-forecast method | |

chemical | variances | Hollingsworth–Lönnberg (HL) method as function of height | Hollingsworth–Lönnberg (HL) method as function of height |

correlations | spatially uncorrelated | Six-hour difference (CQC) method |

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**MDPI and ACS Style**

Ménard, R.; Gauthier, P.; Rochon, Y.; Robichaud, A.; de Grandpré, J.; Yang, Y.; Charrette, C.; Chabrillat, S. Coupled Stratospheric Chemistry–Meteorology Data Assimilation. Part II: Weak and Strong Coupling. *Atmosphere* **2019**, *10*, 798.
https://doi.org/10.3390/atmos10120798

**AMA Style**

Ménard R, Gauthier P, Rochon Y, Robichaud A, de Grandpré J, Yang Y, Charrette C, Chabrillat S. Coupled Stratospheric Chemistry–Meteorology Data Assimilation. Part II: Weak and Strong Coupling. *Atmosphere*. 2019; 10(12):798.
https://doi.org/10.3390/atmos10120798

**Chicago/Turabian Style**

Ménard, Richard, Pierre Gauthier, Yves Rochon, Alain Robichaud, Jean de Grandpré, Yan Yang, Cécilien Charrette, and Simon Chabrillat. 2019. "Coupled Stratospheric Chemistry–Meteorology Data Assimilation. Part II: Weak and Strong Coupling" *Atmosphere* 10, no. 12: 798.
https://doi.org/10.3390/atmos10120798