Studies of Variational Assimilation for the Inversion of the Coupled Air-sea Model

For the prediction of ENSO, the accuracy of the model including the parameters, initial value and others of the model is important, which can be retrieved by the variational data assimilation methods developed in recent years. However, when the nonlinearity of the model is quite strong, the effect of the improvement made by the 4-D variational data assimilation may be poor due to the bad approximation of the tangent linear model to the original model. So in the paper the ideas in the optimal control is introduced to improve the effect of 4-DVAR in the inversion of the parameters of a nonlinear dynamic ENSO model. The results indicate that when the terminal controlling term is added to the cost functional of 4DVAR, which originated from the optimal control, the effect of the inversion may be largely improved comparing to the traditional 4DVAR, as can be especially obvious from the phase orbit of the model variables. The results in the paper also suggest that the method of 4DVAR in combination with optimal control cannot only reduce the error resulting from the inaccuracy of the model parameters but also can correct the parameters itself. This gives a good method in modifying the model and improving the quality of prediction of ENSO.

Evaluation of Two Momentum Control Variable Schemes and Their Impact on the Variational Assimilation of Radar Wind Data:Case Study of a Squall Line

Different choices of control variables in variational assimilation can bring about different influences on the analyzed atmospheric state. Based on the WRF model＇s three-dimensional variational assimilation system, this study compares the be- havior of two momentum control variable options-streamfunction velocity potential （ψ-χ） and horizontal wind components （U-V）-in radar wind data assimilation for a squall line case that occurred in Jiangsu Province on 24 August 2014. The wind increment from the single observation test shows that the ψ-χ control variable scheme produces negative increments in the neighborhood around the observation point because streamfunction and velocity potential preserve integrals of velocity. On the contrary, the U-V control variable scheme objectively reflects the information of the observation itself. Furthermore, radial velocity data from 17 Doppler radars in eastern China are assimilated. As compared to the impact of conventional observation, the assimilation of radar radial velocity based on the U-V control variable scheme significantly improves the mesoscale dynamic field in the initial condition. The enhanced low-level jet stream, water vapor convergence and low-level wind shear result in better squall line forecasting. However, the ψ-χ control variable scheme generates a discontinuous wind field and unrealistic convergence/divergence in the analyzed field, which lead to a degraded precipitation forecast.

Variational Assimilation of GPS Precipitable Water Vapor and Hourly Rainfall Observations for a Meso-βScale Heavy Precipitation Event During the 2002 Mei-Yu Season

Recent advances in Global Positioning System （GPS） remote sensing technology allow for a direct estimation of the precipitable water vapor （PWV） from delayed signals transmitted by GPS satellites, which can be assimilated into numerical models with four-dimensional variational （4DVAR） data assimilation. A mesoscale model and its 4DVAR system are used to access the impacts of assimilating GPS-PWV and hourly rainfall observations on the short-range prediction of a heavy rainfall event on 20 June 2002. The heavy precipitation was induced by a sequence of meso-β-scale convective systems （MCS） along the mei-yu front in China. The experiments with GPS-PWV assimilation cluster and also eliminated the erroneous rainfall successfully simulated the evolution of the observed MCS systems found in the experiment without 4DVAR assimilation. Experiments with hourly rainfall assimilation performed similarly both on the prediction of MCS initiation and the elimination of erroneous systems, however the MCS dissipated much sooner than it did in observations. It is found that the assimilation-induced moisture perturbation and mesoscale low-level jet are helpful for the MCS generation and development. It is also discovered that spurious gravity waves may post serious limitations for the current 4DVAR algorithm, which would degrade the assimilation efficiency, especially for rainfall data. Sensitivity experiments with different observations, assimilation windows and observation weightings suggest that assimilating GPS-PWV can be quite effective, even with the assimilation window as short as 1 h. On the other hand, assimilating rainfall observations requires extreme cautions on the selection of observation weightings and the control of spurious gravity waves.

Variational assimilation in combination with a regularization method for sea level pressure retrieval from QuikSCAT scatterometer data Ⅱ:simulation experiment and actual case study

The sea level pressure field can be computed from sea surface winds retrieved from satellite microwave scatterometer measurements, based on variational assimilation in combination with a regularization method given in part I of this paper. First, the validity of the new method is proved with a simulation experiment. Then, a new processing procedure for the sea level pressure retrieval is built by combining the geostrophic wind, which is computed from the scatterometer 10-meter wind using the University of Washington planetary boundary layer model using this method. Finally, the feasibility of the method is proved using an actual case study.

Variational assimilation in combination with the regularization method for sea level pressure retrieval from QuikSCAT scatterometer data I:Theoretical frame construction

A new method of constructing a sea level pressure field from satellite microwave scatterometer measurements is presented. It is based on variational assimilation in combination with a regularization method using geostrophic vorticity to construct a sea level pressure field from scatterometer data that are given in this paper, which offers a new idea for the application of scatterometer measurements. Firstly, the geostrophic vorticity from the scatterometer data is computed to construct the observation field, and the vorticity field in an area and the sea level pressure on the borders are assimilated. Secondly, the gradient of sea level pressure （semi-norm） is used as the stable functional to educe the adjoint system, the adjoint boundary condition and the gradient of the cost functional in which a weight parameter is introduced for the harmony of the system and the Tikhonov regularization techniques in inverse problem are used to overcome the ill-posedness of the assimilation. Finally, the iteration method of the sea level pressure field is developed.

The Variational Assimilation Experiment of GPS Bending Angle

More and more new types of observational data provide many new opportunities for improving numerical weather forecasts. Among these, the GPS (Global Positioning System) bending angle is undoubtedly very important. There are many advantages of the GPS bending angle, such as high resolution, availability in all weather conditions, and global data coverage. Thus it is very valuable to assimilate GPS bending angle data into numerical weather models. This paper introduces how to obtain and assimilate the GPS bending angle. There are two methods of assimilation: the indirect method and direct method, and they are both introduced in this paper. During the minimizing process of variational assimilation, calculation efficiency is very important and the optimal step size greatly influences the algorithm efficiency. Based on the characteristics of the minimizing algorithm, we obtain an adaptive method for calculating the optimizing step suitable for all kinds of minimization algorithms through mathematical deduction. Finally, a numerical variational assimilation experiment is performed using the GPS bending angle data of 11 October 1995. The numerical results indicate the validity of the variational assimilation method and the adaptive method introduced here.

Variational Assimilation of Satellite Cloud Water/Ice Path and Microphysics Scheme Sensitivity to the Assimilation of a Rainfall Case

Hydrometeor variables （cloud water and cloud ice mixing ratios） are added into the WRF three-dimensional variational assimilation system as additional control variables to directly analyze hydrometeors by assimilating cloud observations. In addition, the background error covariance matrix of hydrometeors is modeled through a control variable transform, and its characteristics discussed in detail. A suite of experiments using four microphysics schemes （LIN, SBU-YLIN, WDM6 and WSM6） are performed with and without assimilating satellite cloud liquid/ice water path. We find analysis of hydrometeors with cloud assimilation to be significantly improved, and the increment and distribution of hydrometeors are consistent with the characteristics of background error covariance. Diagnostic results suggest that the forecast with cloud assimilation represents a significant improvement, especially the ability to forecast precipitation in the first seven hours. It is also found that the largest improvement occurs in the experiment using the WDM6 scheme, since the assimilated cloud information can sustain for longer in this scheme. The least improvement, meanwhile, appears in the experiment using the SBU-YLIN scheme.

Study and application of an improved four-dimensional variational assimilation system based on the physical-space statistical analysis for the South China Sea

The four-dimensional variational assimilation(4D-Var)has been widely used in meteorological and oceanographic data assimilation.This method is usually implemented in the model space,known as primal approach(P4D-Var).Alternatively,physical space analysis system(4D-PSAS)is proposed to reduce the computation cost,in which the 4D-Var problem is solved in physical space(i.e.,observation space).In this study,the conjugate gradient(CG)algorithm,implemented in the 4D-PSAS system is evaluated and it is found that the non-monotonic change of the gradient norm of 4D-PSAS cost function causes artificial oscillations of cost function in the iteration process.The reason of non-monotonic variation of gradient norm in 4D-PSAS is then analyzed.In order to overcome the non-monotonic variation of gradient norm,a new algorithm,Minimum Residual(MINRES)algorithm,is implemented in the process of assimilation iteration in this study.Our experimental results show that the improved 4D-PSAS with the MINRES algorithm guarantees the monotonic reduction of gradient norm of cost function,greatly improves the convergence properties of 4D-PSAS as well,and significantly restrains the numerical noises associated with the traditional 4D-PSAS system.

Four-Dimensional Variational Data Assimilation Experiments for a Heavy Rain Case During the 2002 IOP in China

A heavy rainfall event along the mei-yu front during 22-23 June 2002 was chosen for this study. To assess the impact of the routine and additional IOP (intensive observation period) radiosonde observations on the mesoscale heavy rainfall forecast, a series of four-dimensional variational (4DVAR) data assimilation and model simulation experiments was conducted using nonhydrostatic mesoscale model MM5 and the MM5 4DVAR system. The effects of the intensive observations in the different areas on the heavy rainfall forecast were also investigated. The results showed that improvement of the forecast skill for mesoscale heavy rainfall intensity was possible from the assimilation of the IOP radiosonde observations. However, the impact of the IOP observations on the forecast of the rainfall pattern was not significant. Initial conditions obtained through the 4DVAR experiments with a 12-h assimilation window were capable of improving the 24-h forecast. The simulated results after the assimilation showed that it would be best to perform the intensive radiosonde observations in the upstream of the rainfall area and in the moisture passageway area at the same time. Initial conditions created by the 4DVAR led to the low-level moisture convergence over the rainfall area, enhanced frontogenesis and upward motion within the mei-yu front, and intensified middle- and high-level unstable stratification in front of the mei-yu front. Consequently, the heavy rainfall forecast was improved.

Study on the Variational Assimilation Technique for the Retrieval of Wind Fields from Doppler Radar Data

This paper introduces a variational assimilation technique for the retrieval of wind fields from Doppler radar data. The assimilated information included both the radial velocity （RV） and the movement of radar echo. In this assimilation technique, the key is transforming the movement of radar echo to a new radar measuring variable- ＂apparent velocity＂ （AV）. Thus, the information of wind is added, and the indeterminacy of recovering two-dimensional wind only by AV was overcome effectively by combining RV with AV. By means of CMA GRAPES-3Dvar and CINRAD data, some experiments were performed. The results show that the method of retrieval of wind fields is useful in obtaining the construction of the weather system.

Application of a Recursive Filter to a Three-Dimensional Variational Ocean Data Assimilation System

In order to improve the efficiency of the Ocean Variational Assimilation System （OVALS）, which has been widely used in various applications, an improved OVALS （OVALS2） is developed based on the recursive filter （RF） algorithm. The first advantage of OVALS2 is that memory storage can be substantially reduced in practice because it implicitly computes the background error covariance matrix; the second advantage is that there is no inversion of the background error covariance by preconditioning the control variable. For comparing the effectiveness between OVALS2 and OVALS, a set of experiments was implemented by assimilating expendable bathythermograph （XBT） and ARGO data into the Tropical Pacific circulation model. The results show that the efficiency of OVALS2 is much higher than that of OVALS. The computational time and the computer storage in the assimilation process were reduced by 83% and 77%, respectively. Additionally, the corresponding results produced by the RF are almost as good as those obtained by OVALS. These results prove that OVALS2 is suitable for operational numerical oceanic forecasting.

Coupling Ensemble Kalman Filter with Four-dimensional Variational Data Assimilation

This study examines the performance of coupling the deterministic four-dimensional variational assimilation system （4DVAR） with an ensemble Kalman filter （EnKF） to produce a superior hybrid approach for data assimilation. The coupled assimilation scheme （E4DVAR） benefits from using the state-dependent uncertainty provided by EnKF while taking advantage of 4DVAR in preventing filter divergence： the 4DVAR analysis produces posterior maximum likelihood solutions through minimization of a cost function about which the ensemble perturbations are transformed, and the resulting ensemble analysis can be propagated forward both for the next assimilation cycle and as a basis for ensemble forecasting. The feasibility and effectiveness of this coupled approach are demonstrated in an idealized model with simulated observations. It is found that the E4DVAR is capable of outperforming both 4DVAR and the EnKF under both perfect- and imperfect-model scenarios. The performance of the coupled scheme is also less sensitive to either the ensemble size or the assimilation window length than those for standard EnKF or 4DVAR implementations.

Theoretical analyses and numerical experiments of variational assimilation for one-dimensional ocean temperature model with techniques in inverse problems

In the present work, the data assimilation problem in meteorology and physical oceanography is re-examined using the variational optimal control approaches in combination with regularization techniques in inverse problem. Here the estimations of the initial condition, boundary condition and model parameters are performed simultaneously in the framework of variational data assimilation. To overcome the difficulty of ill-posedness, especially for the model parameters distributed in space and time, an additional term is added into the cost functional as a stabilized functional. Numerical experiments show that even with noisy observations the initial conditions and model parameters are recovered to an acceptable degree of accuracy.

Testing a Four-Dimensional Variational Data Assimilation Method Using an Improved Intermediate Coupled Model for ENSO Analysis and Prediction

A four-dimensional variational （4D-Var） data assimilation method is implemented in an improved intermediate coupled model （ICM） of the tropical Pacific. A twin experiment is designed to evaluate the impact of the 4D-Var data assimilation algorithm on ENSO analysis and prediction based on the ICM. The model error is assumed to arise only from the parameter uncertainty. The ＂observation＂ of the SST anomaly, which is sampled from a ＂truth＂ model simulation that takes default parameter values and has Gaussian noise added, is directly assimilated into the assimilation model with its parameters set erroneously. Results show that 4D-Var effectively reduces the error of ENSO analysis and therefore improves the prediction skill of ENSO events compared with the non-assimilation case. These results provide a promising way for the ICM to achieve better real-time ENSO prediction.

Initialization and Simulation of a Typhoon Using 4-Dimensional Variational Data Assimilation——Research on Typhoon Herb (1996)

The MM5 and its four dimensional variational data assimilation (4D-Var) system are used in this paper. Based on the National Centers for Environmental Prediction/National Center for Atmospheric Research (NCEP/NCAR) reanalysis data, the authors generate an optimal initial condition for a typhoon by using the bogus data assimilation (BDA) scheme. BDA is able to recover many of the structural features of typhoons including a warm-core vertex, the correct center position, and the strong circulation. As a result of BDA using a bogus surface low, dramatic improvement is achieved in the 72 h prediction of typhoon Herb. Through several cases, the initialization by BDA effectively generates the harmonious inner structure of the typhoon, but which is lacking in the original analysis field. Therefore the intensity forecast is improved greatly. Some improvements are made in the track forecast, but more work still needs to be done.

Optimal estimation of zonal velocity and transport through Luzon Strait using variational data assimilation technique

A P-vector method was optimized using variational data assimilation technique, with which the vertical structures and seasonal variations of zonal velocities and transports were investigated. The results showed that westward and eastward flowes occur in the Luzon Strait in the same period in a year. However the net volume transport is westward. In the upper level (0m -500m),the westward flow exits in the middle and south of the Luzon Strait, and the eastward flow exits in the north. There are two centers of westward flow and one center of eastward flow. In the middle of the Luzon Strait, westward and eastward flowes appear alternately in vertical direction. The westward flow strengthens in winter and weakens in summer. The net volume transport is strong in winter (5.53 Sv) but weak in summer (0.29 Sv). Except in summer, the volume transport in the upper level accounts for more than half of the total volume transport (0m bottom). In summer, the net volume transport in the upper level is eastward (1.01 Sv), but westward underneath.

VARIATIONAL DATA ASSIMILATION USING WAVELET BACKGROUND ERROR COVARIANCE: INITIALIZATION OF TYPHOON KAEMI (2006)

Background error covariance plays an important role in any variational data assimilation system, because it determines how information from observations is spread in model space and between different model variables. In this paper, the use of orthogonal wavelets in representation of background error covariance over a limited area is studied. Based on the WRF model and its 3D-VAR system, an algorithm using orthogonal wavelets to model background error covariance is developed. Because each wavelet function contains information on both position and scale, using a diagonal correlation matrix in wavelet space gives the possibility to represent some anisotropic and inhomogeneous characteristics of background error covariance. The experiments show that local correlation functions are better modeled than spectral methods. The formulation of wavelet background error covariance is tested with the typhoon Kaemi (2006). The results of experiments indicate that the subsequent forecasts of typhoon Kaemi’s track and intensity are significantly improved by the new method.

Parallel Computing of a Variational Data Assimilation Model for GPS/MET Observation Using the Ray-Tracing Method

The Spectral Statistical Interpolation (SSI) analysis system of NCEP is used to assimilate meteorological data from the Global Positioning Satellite System (GPS/MET) refraction angles with the variational technique. Verified by radiosonde, including GPS/MET observations into the analysis makes an overall improvement to the analysis variables of temperature, winds, and water vapor. However, the variational model with the ray-tracing method is quite expensive for numerical weather prediction and climate research. For example, about 4 000 GPS/MET refraction angles need to be assimilated to produce an ideal global analysis. Just one iteration of minimization will take more than 24 hours CPU time on the NCEP's Cray C90 computer. Although efforts have been taken to reduce the computational cost, it is still prohibitive for operational data assimilation. In this paper, a parallel version of the three-dimensional variational data assimilation model of GPS/MET occultation measurement suitable for massive parallel processors architectures is developed. The divide-and-conquer strategy is used to achieve parallelism and is implemented by message passing. The authors present the principles for the code's design and examine the performance on the state-of-the-art parallel computers in China. The results show that this parallel model scales favorably as the number of processors is increased. With the Memory-IO technique implemented by the author, the wall clock time per iteration used for assimilating 1420 refraction angles is reduced from 45 s to 12 s using 1420 processors. This suggests that the new parallelized code has the potential to be useful in numerical weather prediction (NWP) and climate studies.

A Prototype Regional GSI-based EnKF-Variational Hybrid Data Assimilation System for the Rapid Refresh Forecasting System：Dual-Resolution Implementation and Testing Results

A dual-resolution（DR） version of a regional ensemble Kalman filter（EnKF）-3D ensemble variational（3DEnVar） coupled hybrid data assimilation system is implemented as a prototype for the operational Rapid Refresh forecasting system. The DR 3DEnVar system combines a high-resolution（HR） deterministic background forecast with lower-resolution（LR） EnKF ensemble perturbations used for flow-dependent background error covariance to produce a HR analysis. The computational cost is substantially reduced by running the ensemble forecasts and EnKF analyses at LR. The DR 3DEnVar system is tested with 3-h cycles over a 9-day period using a 40/13-km grid spacing combination. The HR forecasts from the DR hybrid analyses are compared with forecasts launched from HR Gridpoint Statistical Interpolation（GSI） 3D variational（3DVar）analyses, and single LR hybrid analyses interpolated to the HR grid. With the DR 3DEnVar system, a 90% weight for the ensemble covariance yields the lowest forecast errors and the DR hybrid system clearly outperforms the HR GSI 3DVar.Humidity and wind forecasts are also better than those launched from interpolated LR hybrid analyses, but the temperature forecasts are slightly worse. The humidity forecasts are improved most. For precipitation forecasts, the DR 3DEnVar always outperforms HR GSI 3DVar. It also outperforms the LR 3DEnVar, except for the initial forecast period and lower thresholds.

On the 4D Variational Data Assimilation with Constraint Conditions

An investigation is carried out on the problem involved in 4D variational data assimilation (VDA) with constraint conditions based on a finite-element shallow-water equation model. In the investigation, the adjoint technology, penalty method and augmented Lagrangian method are used in constraint optimization field to minimize the defined constraint objective functions. The results of the numerical experiments show that the optimal solutions are obtained if the functions reach the minima. VDA with constraint conditions controlling the growth of gravity oscillations is efficient to eliminate perturbation and produces optimal initial field. It seems that this method can also be applied to the problem in numerical weather prediction. Key words Variational data assimilation - Constraint conditions - Penalty methods - finite-element model This research is supported by National Natural Science Foundation of China (Grant No. 49575269) and by National Key Basic Research on the Formation Mechanism and Prediction Theory of Severe Synoptic Disasters (Grant No. G1998040910).