Impacts of Tangent Linear Model Errors in 4DVar Using a Global Shallow Water Model

A four-dimensional variational data assimilation (4DVar) system of the LASG/IAP Climate Ocean Model, version 1.0 (LICOM1.0), named LICOM-3DVM, has been developed using the three-dimensional variational data assimilation of mapped observation (3DVM), a 4DVar method newly proposed in the past two years. Two experiments with 12-year model integrations were designed to validate it. One is the assimilation run, called ASSM, which incorporated the analyzed weekly sea surface temperature (SST) fields from Reynolds and Smith (OISST) between 1990 and 2001 once a week by the LICOM-3DVM. The other is the control run without any assimilation, named CTL. ASSM shows that the simulated temperatures of the upper ocean (above 50 meters), especially the SST of equatorial Pacific, coincide with the Tropic Atmosphere Ocean (TAO) mooring data, the World Ocean Atlas 2001 (WOA01) data and the Met Office Hadley Centre’s sea ice and sea surface temperature (HadISST) data. It decreased the cold bias existing in CTL in the eastern Pacific and produced a Nino index that agrees with observation well. The validation results suggest that the LICOM-3DVM is able to effectively adjust the model results of the ocean temperature, although it’s hard to correct the subsurface results and it even makes them worse in some areas due to the incorporation of only surface data. Future development of the LICOM-3DVM is to include subsurface in situ observations and satellite observations to further improve model simulations.

The mainstream data assimilation system in operation will still employ four-dimensional variational data assimilation(4DVAR) method in a long time of future. We develop a new 4DVAR system, i.e., YH4DVAR, using global spectral model as a constraint to impose a dynamic balance on the assimilation. The cost function of YH4DVAR consists of four terms: background, observations, digital filter, and bias correction, respectively. YH4DVAR employs the wavelet background error convariance, the multi-resolution incremental formulation, the tangent linear and adjoint models for dynamics core and physical processes, and ATOVS radiance data assimilation. Simultaneous spatial and spectral variations of horizontal and vertical covariance are achieved by dividing the control vector into several parts, each of which corresponds to a band of total spherical wavenumbers. Using NWP consisting of YH4DVAR and global spectral model, the anomaly correlation with the verifying analysis for geopotential height 8-day forecast on the 500 hPa isobaric surface at 12-month mean is above 60%. The formulation and implementation of YH4VAR are described in detail in this paper.

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.

The use of Four-Dimensional variational (4D-Var) data assimilation technology in the context of sea dynamics problems, with a sensitivity analysis of model results to observation errors, is presented. The technology is applied to a numerical model of ocean circulation developed at the Marchuk Institute of Numerical Mathematics, Russian Academy of Sciences (INM RAS), with the use of the splitting method and complemented by 4D-Var data assimilation with covariance matrices of background and observation errors. The variational data assimilation involves iterative procedures to solve inverse problems so as to correct sea surface heat fluxes for the model under consideration. An algorithm is formulated to study the sensitivity of the model outputs, considered as output functions after assimilation, to the observation errors. The algorithm reveals the regions where the output function gradient is the largest for the average sea surface temperature (SST) in a selected area, obtained by assimilation. In the numerical experiments, a 4D variational problem of SST assimilation for the Baltic Sea area is solved.

Abstract. We have examined the utility of retrieved column-averaged, dry-air mole fractions of CO2 (XCO2) from the Greenhouse Gases Observing Satellite (GOSAT) for quantifying monthly, regional flux estimates of CO2, using the GEOS-Chem four-dimensional variational (4D-Var) data assimilation system. We focused on assessing the potential impact of biases in the GOSAT CO2 data on the regional flux estimates. Using different screening and bias correction approaches, we selected three different subsets of the GOSAT XCO2 data for the 4D-Var inversion analyses, and found that the inferred global fluxes were consistent across the three XCO2 inversions. However, the GOSAT observational coverage was a challenge for the regional flux estimates. In the northern extratropics, the inversions were more sensitive to North American fluxes than to European and Asian fluxes due to the lack of observations over Eurasia in winter and over eastern and southern Asia in summer. The regional flux estimates were also sensitive to the treatment of the residual bias in the GOSAT XCO2 data. The largest differences obtained were for temperate North America and temperate South America, for which the largest spread between the inversions was 1.02 and 0.96 Pg C, respectively. In the case of temperate North America, one inversion suggested a strong source, whereas the second and third XCO2 inversions produced a weak and strong sink, respectively. Despite the discrepancies in the regional flux estimates between the three XCO2 inversions, the a posteriori CO2 distributions were in good agreement (with a mean difference between the three inversions of typically less than 0.5 ppm) with independent data from the Total Carbon Column Observing Network (TCCON), the surface flask network, and from the HIAPER Pole-to-Pole Observations (HIPPO) aircraft campaign. The discrepancy in the regional flux estimates from the different inversions, despite the agreement of the global flux estimates suggests the need for additional work to determine the minimum spatial scales at which we can reliably quantify the fluxes using GOSAT XCO2. The fact that the a posteriori CO2 from the different inversions were in good agreement with the independent data although the regional flux estimates differed significantly, suggests that innovative ways of exploiting existing data sets, and possibly additional observations, are needed to better evaluate the inferred regional flux estimates.

The four-dimensional variational (4D-Var) data assimilation systems used in most operational and research centers use initial condition increments as control variables and adjust initial increments to find optimal analysis solutions. This approach may sometimes create discontinuities in analysis fields and produce undesirable spin ups and spin downs. This study explores using incremental analysis updates (IAU) in 4D-Var to reduce the analysis discontinuities. IAU-based 4D-Var has almost the same mathematical formula as conventional 4D-Var if the initial condition increments are replaced with time-integrated increments as control variables.

Pertaining the application of the 4DVar and 4DEnVar WRFDA techniques to simulate tropical cyclones in the Bay of Bengal

The ensemble-based tangent linear model (ETLM), which evolves the static background error covariance (BEC) in time in the data assimilation window, is beneficial for improving the analysis and avoids the need for developing the tangent linear forecast model and its adjoint as in four-dimensional variational data assimilation (4D-Var). This study proposes to apply the filtered ETLM (FETLM), which is composed of ensemble perturbations at limited time slots and localized with a quasi-Gaussian filter in the way that avoids calculation of the inverse matrix. Tests with the Lorenz model showed that FETLM evolved the static BEC as in 4D-Var and improved the analysis. In a case study with the FETLM implemented in the hourly updated regional forecast system, the Rapid Refresh Forecast System, the imbalance of the analysis was mitigated due to the flow-dependent BEC. Moreover, the cross-variable covariance in the FETLM made it possible to assimilate radar reflectivity effectively even without hydrometeors as control variables. Significance Statement In atmospheric variational data assimilation, the ensemble-based tangent linear model (ETLM), which evolves the static background error covariance (BEC) in time, is beneficial for improving the analysis. However, it generally requires significant computation with high-frequency ensemble forecasts. This study proposes a filtered ETLM (FETLM) to avoid the huge computation. Tests with an hourly updated regional forecast system showed that FETLM mitigated the imbalance of the analysis and made it possible to assimilate radar reflectivity more effectively due to the flow-dependent BEC.

This chapter reviews developments in representer-based variational data assimilation systems over the past 15 years. Data assimilation systems with representer-based algorithms are routinely used in operational and research centers for producing four-dimensional atmospheric and oceanic analyses and prediction. The systems reviewed in this chapter are the Inverse Ocean Modeling (IOM) system, the Naval Research Laboratory Atmospheric Variational Data Assimilation System-Accelerated Representer (NAVDAS-AR) system, the Navy Coastal Ocean Model 4D-Var (NCOM 4D-Var) system, and the Regional Ocean Modeling System 4D-Var (ROMS 4D-Var ) system. These systems are mature operational or semi-operational weak-constraint, four-dimensional variational data assimilation systems. The emphasis here is on providing brief reviews with the key references related to the implementation and applications of these systems. Readers interested in early developments of representer-based systems (before 2002) are encouraged to look at Chua and Bennett (2001) and Bennett (2002).

An observing system experiment, simulating a surface-only observing network representative of the 1930s, is carried out with three- and four-dimensional variational data assimilation systems (3D-VAR and 4D-VAR) and an ensemble-based data assimilation system (EnsDA). It is found that 4D-VAR and EnsDA systems produce analyses of comparable quality and that both are much more accurate than the analyses produced by the 3D-VAR system. The EnsDA system also produces useful estimates of analysis error, which are not directly available from the variational systems.

ABSTRACTFour-dimensional variational (4D-Var) data assimilation for operational systems requires the solution of large linear systems that are poorly conditioned in general. In addition to efficient iterative solvers for linear systems, using a good preconditioner is required to guarantee an acceptable solution with a small number of iterations. We consider the assimilation of ocean observations using a weak constraints 4D-Var for the Navy Coastal Ocean Model (NCOM), based on the representer method. Two methods of preconditioning the linear system are implemented, namely the scaling of the linear system by the square root of the observations error variances, and the approximating stabilised representer matrix based on the computation of some representer functions. We evaluate their convergence using criteria such as the norm of the residuals and of the gradient of the cost function, the analysis error evaluated at the observation locations, and finally the convergence of the sequence of analyses. Results from all criteria show that the fastest convergence is achieved with the rescaling preconditioner when the conjugate gradient used to solve the linear system is equipped with a suitable inner product instead of the Euclidean.

Predictive skill comparative assessment of WRF 4DVar and 3DVar data assimilation: An Indian Ocean tropical cyclone case study

Abstract. The wind extraction due to assimilation of stratospheric trace gas (tracer) data is examined using a 4D-Var (four-dimensional variational) data assimilation system based on the shallow water equations coupled to the tracer continuity equation. The procedure is outlined as follows. First, a nature run is created, simulating middle stratospheric winter conditions. Second, ozone (O3), nitrous oxide (N2O), and water vapor (H2O) (treated in this study as passive tracers) are initialized using Aura Microwave Limb Sounder (MLS) mixing ratios at 850 K potential temperature and are advected by the nature run winds. Third, the initial dynamical conditions are perturbed by using a 6 h offset. Fourth, simulated hourly tracer observations on the full model grid are assimilated with a 4D-Var system in which tracer and winds are coupled via the adjoint of the tracer continuity equation. Multiple assimilation experiments are performed by varying the amount of random observation error added to the simulated measurements. Finally, the wind extraction potential (WEP) is calculated as the reduction of the vector wind root mean square error (RMSE) relative to the maximum possible reduction. For a single 6 h assimilation cycle with the smallest observation error, WEP values are ~60% for all three tracers, while 10-day multi-cycle simulations result in WEP of ~90%, wind errors of ~0.3 m s−1, and height errors of ~13 m. There is therefore sufficient information in the tracer fields to almost completely constrain the dynamics, even without direct assimilation of dynamical information. When realistic observation error is added (based on MLS precisions at 10 hPa), the WEP after 10 days is 90% for O3, 87% for N2O, and 72% for H2O. O3 and N2O provide more wind information than H2O due to stronger background gradients relative to the MLS precisions. The RMSE for wind reach a minimum level of ~0.3–0.9 m s−1 for the MLS precisions, suggesting a limit to which realistic tracers could constrain the winds, given complete global cover age. With higher observation noise levels, the WEP values decrease, but the impact on the winds is still positive up to noise levels of 100% (relative to the global mean value) when compared to the case of no data assimilation.

Abstract. This paper presents a three-dimensional variational (3DVAR) data assimilation (DA) system for aerosol optical properties, including aerosol optical thickness (AOT) retrievals and lidar-based aerosol profiles, developed for the Model for Simulating Aerosol Interactions and Chemistry (MOSAIC) within the Weather Research and Forecasting model coupled to Chemistry (WRF-Chem) model. For computational efficiency, 32 model variables in the MOSAIC_4bin scheme are lumped into 20 aerosol state variables that are representative of mass concentrations in the DA system. To directly assimilate aerosol optical properties, an observation operator based on the Mie scattering theory was employed, which was obtained by simplifying the optical module in WRF-Chem. The tangent linear (TL) and adjoint (AD) operators were then established and passed the TL/AD sensitivity test. The Himawari-8 derived AOT data were assimilated to validate the system and investigate the effects of assimilation on both AOT and PM2.5 simulations. Two comparative experiments were performed with a cycle of 24 h from 23 to 29 November 2018, during which a heavy air pollution event occurred in northern China. The DA performances of the model simulation were evaluated against independent aerosol observations, including the Aerosol Robotic Network (AERONET) AOT and surface PM2.5 measurements. The results show that Himawari-8 AOT assimilation can significantly improve model AOT analyses and forecasts. Generally, the control experiments without assimilation seriously underestimated AOTs compared with observed values and were therefore unable to describe real aerosol pollution. The analysis fields closer to observations improved AOT simulations, indicating that the system successfully assimilated AOT observations into the model. In terms of statistical metrics, assimilating Himawari-8 AOTs only limitedly improved PM2.5 analyses in the inner simulation domain (D02); however, the positive effect can last for over 24 h. Assimilation effectively enlarged the underestimated PM2.5 concentrations to be closer to the real distribution in northern China, which is of great value for studying heavy air pollution events.

The Indian Space Research Organization launched SCATterometer Satellite‐1 (SCATSat­1) satellite by PSLV‐C35 on 26 September 2016 from Sriharikota. In this study, SCATSat‐1 retrieved winds are assimilated in the regional atmospheric model subsequent to preliminary evaluation of SCATSat‐1 retrieved winds against National Centers for Environmental Prediction final analyses. The Weather Research and Forecasting (WRF) model and its three‐ and four‐dimensional variational (3D‐Var and 4D‐Var) data assimilation system are used to demonstrate the importance of SCATSat‐1 retrieved winds on short‐range weather prediction. In the first set of experiments, two parallel runs are performed with and without assimilation of SCATSat‐1 retrieved winds using 6‐hourly cycling during entire month of December 2016. Results suggest that assimilation of SCATSat‐1 retrieved winds improved the WRF model analyses and subsequent forecasts of winds and moisture. In the second set of experiments, parallel assimilation experiments are performed with SCATSat‐1 retrieved winds using 3D‐Var and 4D‐Var method for the entire lifetime of tropical cyclone Ockhi as a case study. Results suggest that assimilation of SCATSat‐1 winds has positive impact on Ockhi tropical cyclone prediction with slightly better forecast skill for 4D‐Var technique. Moreover, assimilation of SCATSat‐1 retrieved winds improved predictions of surface convergence and accumulated rainfall. In the last set of experiments, forecast sensitivity to observations are performed for SCATSat‐1 retrieved different resolution winds using the WRF tangent linear and adjoint models for the entire lifetime of tropical cyclone Ockhi. Results suggest that SCATSat‐1 retrieved winds have positive impact on tropical cyclone simulation.