Minimization algorithms are singular components in four-dimensional variational data assimilation (4DVar). In this paper, the convergence and application of the conjugate gradient algorithm (CGA), which is based on the Lanczos iterative algorithm and the Hessian matrix derived from tangent linear and adjoint models using a non-hydrostatic framework, are investigated in the 4DVar minimization. First, the influence of the Gram-Schmidt orthogonalization of the Lanczos vector on the convergence of the Lanczos algorithm is studied. The results show that the Lanczos algorithm without orthogonalization fails to converge after the ninth iteration in the 4DVar minimization, while the orthogonalized Lanczos algorithm converges stably. Second, the convergence and computational efficiency of the CGA and quasi-Newton method in batch cycling assimilation experiments are compared on the 4DVar platform of the Global/Regional Assimilation and Prediction System (GRAPES). The CGA is 40% more computationally efficient than the quasi-Newton method, although the equivalent analysis results can be obtained by using either the CGA or the quasi-Newton method. Thus, the CGA based on Lanczos iterations is better for solving the optimization problems in the GRAPES 4DVar system.

Tropical cyclones (TCs) can cause severe wind and rain hazards. Unusual TC tracks and their extreme precipitation forecasts have become two difficult problems faced by conventional models of primitive equations. The case study in this paper finds that the numerical computation of the climatological component in conventional models restricts the prediction of unusual TC tracks. The climatological component should be a forcing quantity, not a predictor in the numerical integration of all models. Anomaly-based variable models can overcome the bottleneck of forecast time length or the one-week forecasting barrier, which is limited to less than one week for conventional models. The challenge in extreme precipitation forecasting is how to physically get the vertical velocity. The anomalous moisture stress modulus (AMSM), as an indicator of heavy rainfall presented in this paper, considers the two conditions associated with vertical velocity and anomalous specific humidity in the lower troposphere. Vertical velocity is produced by the orthogonal collision of horizontal anomalous airflows.

Many ensemble-based data assimilation (DA) methods use observation space localization to mitigate the sampling errors due to the insufficient ensemble members. Observation space localization is simpler and more timesaving than model space localization in implementation, but more difficult to directly assimilate satellite radiance observations, a kind of non-local observations. The vertical locations of radiance observations are undetermined and the transmission of observational information is thereby obstructed. To determine the vertical coordinates of radiance observations, a weighted average hypsometry is proposed. Using this hypsometry, AMSU-A radiance observations are directly assimilated with an ensemble four-dimensional variational (En4DVar) DA system. It consists of a four-dimensional ensemble-variational (4DEnVar) system providing ensemble covariance and a 4DVar system. Observing system simulation experiments show that the hypsometry alleviates the degradations in the late period of medium-range forecast in the Northern Extratropics that occur in the traditional peak-based hypsometry. It obviously improves the analysis qualities and forecast skills of the En4DVar system and its two components, especially in the Southern Extratropics, when incorporating AMSU-A radiance observations. The improvement in the En4DVar-initialized forecast is comparable to that in the 4DVar-initialized forecast in the Southern Extratropics and Tropics. It indicates that a proper hypsometry enables efficient extraction of useful information from AMSU-A radiance observations by 4DEnVar with observation space localization. Therefore, the 4DEnVar provides high-quality ensemble covariances for En4DVar.

We developed a strongly coupled chemistry meteorology four-dimensional variational (4D-Var) assimilation system, CMA-GFS-AERO 4D-Var, for investigating the feedbacks of chemical data assimilation on meteorological forecasts. This system was developed on the basis of the framework of the incremental analysis scheme of the China Meteorological Administration Global Forecasting System (CMA-GFS). CMA-GFS-AERO 4D-Var includes three component models: forward, tangent linear, and adjoint models. CMA-GFS-AERO forward model was constructed by integrating an aerosol module containing main physical processes of black carbon (BC) aerosol in the atmosphere into the CMA-GFS weather model. The tangent linear and the adjoint of the aerosol module was further developed and coupled online with the CMA-GFS tangent linear and adjoint models, respectively. In CMA-GFS-AERO 4D-Var, the BC mass concentration was used as the control variable and minimized together with atmospheric variables. The validation of this system includes the tangent linear approximation, the adjoint correctness test, the single-point observation ideal experiment and the full observation experiment. The results show that CMA-GFS-AERO tangent linear model performs well in tangent linear approximation for BC, and adjoint sensitivity agrees well with tangent linear sensitivity. Assimilating BC observations can generate analysis increments not only for BC but also for atmospheric variables, highlighting the capability of CMA-GFS-AERO 4D-Var in exploring the feedback effect of BC assimilation on atmospheric variables. The computational performance of CMA-GFS-AERO 4D-Var also indicates the potential in operational application. This study focuses on the theoretical architecture and practical implementation of the system, the detailed analysis of the batch test will be described in part 2 of this paper.

We developed a strongly coupled aerosol–meteorology four-dimensional variational (4D-Var) assimilation system, CMA-GFS-AERO 4D-Var, for investigating the feedback of aerosol data assimilation on meteorological forecasts. This system was developed on the basis of the framework of the incremental analysis scheme of the China Meteorological Administration Global Forecasting System (CMA-GFS). CMA-GFS-AERO 4D-Var includes three component models: forward, tangent linear, and adjoint models. The CMA-GFS-AERO forward model was constructed by integrating an aerosol module containing the main physical processes of black carbon (BC) aerosol in the atmosphere into the CMA-GFS weather model. The tangent linear model and the adjoint model of the aerosol module were further developed and coupled online with the CMA-GFS tangent linear and adjoint models, respectively. In CMA-GFS-AERO 4D-Var, the BC mass concentration was used as the control variable and minimized together with atmospheric variables. The validation of this system includes the tangent linear approximation, the adjoint correctness test, the single-observation experiment, and the full-observation experiment. The results show that the CMA-GFS-AERO tangent linear model performs well in terms of tangent linear approximation for BC and that adjoint sensitivity agrees well with tangent linear sensitivity. Assimilating BC observations can generate analysis increments not only for BC but also for atmospheric variables, highlighting the capability of CMA-GFS-AERO 4D-Var in exploring the feedback of BC assimilation on atmospheric variables. The computational performance of CMA-GFS-AERO 4D-Var also indicates its potential in operational application. This study focuses on the theoretical architecture and practical implementation of the system; the detailed analysis of a series of cycling assimilation experiments will be described in part 2 of this set of companion papers.