Four-dimensional variational data assimilation for mesoscale and storm-scale applications

Summary¶The status and progress of the four-dimensional variational data assimilation (4DVAR) are briefly reviewed focusing on application to prediction of mesoscale/storm-scale atmospheric phenomena. Theoretical background is provided for each important component of the 4DVAR system – forecast and adjoint models, observations, background, cost function, preconditioning, and minimization. An overview of practical issues specific for mesoscale/storm-scale 4DVAR is then presented in terms of high-resolution observations, nonlinearity and discontinuity problem, model error, errors from lateral boundary condition, and precipitation assimilation. Practical strategies for efficient and simplified 4DVAR are also introduced, e.g., incremental 4DVAR, poor man’s 4DVAR, and inverse 3DVAR. A new concept on hybrid approach is proposed to combine an efficient 4DVAR scheme and the standard 4DVAR scheme aiming at reducing computational demand required by the standard 4DVAR while improving the accuracy of the simplified 4DVAR. Applications to both hydrostatic and nonhydrostatic models are illustrated and our vision on opportunities and directions for future research is provided.

[1]  Ionel M. Navon,et al.  Conjugate-Gradient Methods for Large-Scale Minimization in Meteorology , 1987 .

[2]  J. Thepaut,et al.  Impact of a simplified physical package in 4D‐var analyses of fastex situations , 1999 .

[3]  Jean-Noël Thépaut,et al.  Variational inversion of simulated TOVS radiances using the adjoint technique , 1990 .

[4]  Seon Ki Park,et al.  Validity of the Tangent Linear Approximation in a Moist Convective Cloud Model , 1997 .

[5]  S. Park.  Nonlinearity and predictability of convective rainfall associated with water vapor perturbations in a numerically simulated storm , 1999 .

[6]  Qin Xu.  Generalized Adjoint for Physical Processes with Parameterized Discontinuities. Part III: Multiple Threshold Conditions , 1997 .

[7]  J. Whitaker,et al.  An Adjoint Sensitivity Study of Blocking in a Two-Layer Isentropic Model , 1993 .

[8]  R. James Purser,et al.  Recursive Filter Objective Analysis of Meteorological Fields: Applications to NESDIS Operational Processing , 1995 .

[9]  Ying-Hwa Kuo,et al.  Regional Real-Time Numerical Weather Prediction: Current Status and Future Potential , 1998 .

[10]  Michael Ghil,et al.  Advanced data assimilation in strongly nonlinear dynamical systems , 1994 .

[11]  Ionel M. Navon,et al.  An Optimal Nudging Data Assimilation Scheme Using Parameter Estimation , 1992 .

[12]  Milija Zupanski,et al.  Regional Four-Dimensional Variational Data Assimilation in a Quasi-Operational Forecasting Environment , 1993 .

[13]  Lance M. Leslie,et al.  Tropical Cyclone Prediction Using a Barotropic Model Initialized by a Generalized Inverse Method , 1993 .

[14]  S. Cohn.  Dynamics of Short-Term Univariate Forecast Error Covariances , 1993 .

[15]  K. Browning.  The mesoscale data base and its use in mesoscale forecasting , 1989 .

[16]  J. Derber,et al.  A reformulation of the background error covariance in the ECMWF global data assimilation system , 1999 .

[17]  Qin Xu.  Generalized adjoint for physical processes with parameterized discontinuities. Part II: Vector formulations and matching conditions , 1996 .

[18]  P. Houtekamer,et al.  A Sequential Ensemble Kalman Filter for Atmospheric Data Assimilation , 2001 .

[19]  Juanzhen Sun,et al.  Dynamical and Microphysical Retrieval from Doppler Radar Observations Using a Cloud Model and Its Adjoint. Part II: Retrieval Experiments of an Observed Florida Convective Storm , 1998 .

[20]  Bruno Luong,et al.  A variational method for the resolution of a data assimilation problem in oceanography , 1998 .

[21]  Clifford H. Dey,et al.  Observing-Systems Simulation Experiments: Past, Present, and Future , 1986 .

[22]  Some statistical considerations associated with the data assimilation of precipitation observations , 2000 .

[23]  Arthur Gelb,et al.  Applied Optimal Estimation , 1974 .

[24]  M. Zupanski,et al.  A preconditioning algorithm for large-scale minimization problems , 1993 .

[25]  Tadashi Tsuyuki,et al.  Variational Data Assimilation in the Tropics Using Precipitation Data. Part II: 3D Model , 1996 .

[27]  Nicole Rostaing,et al.  Automatic differentiation in Odyssée , 1993 .

[28]  Andrew C. Lorenc,et al.  Analysis methods for numerical weather prediction , 1986 .

[29]  Istvan Szunyogh,et al.  Sensitivity of Forecast Errors to Initial Conditions with a Quasi-Inverse Linear Method , 1997 .

[30]  Jidong Gao,et al.  A Variational Method for the Analysis of Three-Dimensional Wind Fields from Two Doppler Radars , 1999 .

[31]  N. Gustafsson,et al.  Using an adjoint model to improve an optimum interpolation-based data-assimilation system , 1997 .

[32]  Christian Bischof,et al.  Sensitivity analysis of the MM5 weather model using automatic differentiation , 1996 .

[33]  X. Zou,et al.  Studies on the Initialization and Simulation of a Mature Hurricane Using a Variational Bogus Data Assimilation Scheme , 2000 .

[34]  Chungu Lu,et al.  Four-Dimensional Variational Data Assimilation for Limited-Area Models: Lateral Boundary Conditions, Solution Uniqueness, and Numerical Convergence , 2000 .

[35]  P. Courtier,et al.  Variational assimilation of meteorological observations with the direct and adjoint shallow-water equations , 1990 .

[36]  Sundar A. Christopher,et al.  The GOES I–M Imagers: New Tools for Studying Microphysical Properties of Boundary Layer Stratiform Clouds , 2000 .

[37]  Y. Sasaki.  SOME BASIC FORMALISMS IN NUMERICAL VARIATIONAL ANALYSIS , 1970 .

[38]  M. Ghil,et al.  Data assimilation in meteorology and oceanography , 1991 .

[39]  P. Courtier,et al.  The ECMWF implementation of three‐dimensional variational assimilation (3D‐Var). I: Formulation , 1998 .

[40]  Andreas Griewank,et al.  Evaluating derivatives - principles and techniques of algorithmic differentiation, Second Edition , 2000, Frontiers in applied mathematics.

[41]  Qin Xu.  Generalized Adjoint for Physical Processes with Parameterized Discontinuities. Part I: Basic Issues and Heuristic Examples , 1996 .

[42]  I. Orlanski.  A rational subdivision of scales for atmospheric processes , 1975 .

[43]  Arthur P. Mizzi,et al.  Diabatic initialization for improvement in the tropical analysis of divergence and moisture using satellite radiometric imagery data , 1994 .

[44]  M. King,et al.  Monitoring earth's vital signs. , 2000, Scientific American.

[45]  J. Dudhia,et al.  Four-Dimensional Variational Data Assimilation of Heterogeneous Mesoscale Observations for a Strong Convective Case , 2000 .

[46]  Ying-Hwa Kuo,et al.  Assimilation of Atmospheric Radio Refractivity Using a Nonhydrostatic Adjoint Model , 1995 .

[47]  D. Stauffer,et al.  Use of Four-Dimensional Data Assimilation in a Limited-Area Mesoscale Model. Part I: Experiments with Synoptic-Scale Data , 1990 .

[48]  Adjoint sensitivity of the Earth's radiation budget in the NCEP medium‐range forecasting model , 1998 .

[49]  F. L. Dimet,et al.  Optimal Determination Of Nudging Coefficients , 2003 .

[50]  Ronald M. Errico,et al.  An examination of the accuracy of the linearization of a mesoscale model with moist physics , 1999 .

[51]  Jidong Gao,et al.  Application of the Quasi-Inverse Method to Data Assimilation , 2000 .

[52]  Guy Chavent.  On the theory and practice of non-linear least-squares , 1991 .

[53]  Z. Pu,et al.  Forecast sensitivity with dropwindsonde data and targeted observations , 1998 .

[54]  F. L. Dimet,et al.  Variational algorithms for analysis and assimilation of meteorological observations: theoretical aspects , 1986 .

[55]  Moti Segal,et al.  Influences of Model Parameterization Schemes on the Response of Rainfall to Soil Moisture in the Central United States , 1996 .

[56]  Jean-Noël Thépaut,et al.  Four‐dimensional variational analyses of FASTEX situations using special observations , 1999 .

[57]  Mesoscale Modeling of the Atmosphere , 1994 .

[58]  Olivier Talagrand,et al.  Assimilation of Observations, an Introduction (gtSpecial IssueltData Assimilation in Meteology and Oceanography: Theory and Practice) , 1997 .

[59]  Werner Wergen,et al.  The effect of model errors in variational assimilation , 1992 .

[60]  J. Mahfouf,et al.  The ecmwf operational implementation of four‐dimensional variational assimilation. III: Experimental results and diagnostics with operational configuration , 2000 .

[61]  David J. Stensrud,et al.  Behaviors of Variational and Nudging Assimilation Techniques with a Chaotic Low-Order Model , 1992 .

[62]  Arthur E. Bryson,et al.  Applied Optimal Control , 1969 .

[63]  Charles A. Doswell,et al.  On the Use of Mesoscale and Cloud-Scale Models in Operational Forecasting , 1992 .

[64]  Yong Li.  A note on the uniqueness problem of variational adjustment approach to four-dimensional data assimilation , 1991 .

[65]  Luc Fillion,et al.  Coupling of Moist-Convective and Stratiform Precipitation Processes for Variational Data Assimilation , 2000 .

[66]  Bing Wu,et al.  Dynamical and Microphysical Retrievals from Doppler Radar Observations of a Deep Convective Cloud , 2000 .

[67]  J. Derber.  A Variational Continuous Assimilation Technique , 1989 .

[68]  Ionel M. Navon,et al.  Use of differentiable and nondifferentiable optimization algorithms for variational data assimilation with discontinuous cost functions , 2000 .

[69]  J. J. Moré,et al.  Quasi-Newton Methods, Motivation and Theory , 1974 .

[70]  Ying-Hwa Kuo,et al.  Variational Assimilation of Precipitable Water Using a Nonhydrostatic Mesoscale Adjoint Model. Part I: Moisture Retrieval and Sensitivity Experiments , 1996 .

[71]  W. J. Steenburgh,et al.  Short-Term Forecast Validation of Six Models , 1999 .

[72]  I. M. Navon,et al.  Impact of Parameter Estimation on the Performance of the FSU Global Spectral Model Using Its Full-Physics Adjoint , 1999 .

[73]  L. Berre.  Estimation of Synoptic and Mesoscale Forecast Error Covariances in a Limited-Area Model , 2000 .

[74]  Robert James Purser,et al.  A Bayesian technique for estimating continuously varying statistical parameters of a variational assimilation , 2003 .

[75]  Christian Bischof,et al.  Adifor 2.0: automatic differentiation of Fortran 77 programs , 1996 .

[76]  R. E. Kalman,et al.  A New Approach to Linear Filtering and Prediction Problems , 2002 .

[77]  Luc Fillion,et al.  Variational Assimilation of Precipitation Data Using Moist Convective Parameterization Schemes: A 1D-Var Study , 1997 .

[78]  Dusanka Zupanski,et al.  Fine-Resolution 4DVAR Data Assimilation for the Great Plains Tornado Outbreak of 3 May 1999 , 2002 .

[79]  L. S. Pontryagin,et al.  Mathematical Theory of Optimal Processes , 1962 .

[80]  A. Jazwinski.  Stochastic Processes and Filtering Theory , 1970 .

[81]  D. Stauffer,et al.  Optimal determination of nudging coefficients using the adjoint equations , 1993 .

[82]  Mark DeMaria,et al.  Optimization of a Hurricane Track Forecast Model with the Adjoint Model Equations , 1993 .

[83]  F. Mesinger,et al.  Four-dimensional variational assimilation of precipitation data , 1995 .

[84]  M. Zupanski.  A Preconditioning Algorithm for Four-Dimensional Variational Data Assimilation , 1996 .

[85]  X. Zou,et al.  Introduction to Adjoint Techniques and the MM5 Adjoint Modeling System. , 1997 .

[86]  Roel Snieder.  The role of nonlinearity in inverse problems , 1998 .

[87]  Ionel M. Navon,et al.  Variational data assimilation with moist threshold processes using the NMC spectral model , 1993 .

[88]  Seon Ki Park,et al.  Sensitivity Analysis of a 3D Convective Storm: Implications for Variational Data Assimilation and Forecast Error , 2000 .

[89]  J. Bao,et al.  Treatment of on/off switches in the adjoint method: FDDA experiments with a simple model , 1993 .

[90]  U. Seiler.  Estimation of open boundary conditions with the adjoint method , 1993 .

[91]  Ionel Michael Navon,et al.  Performance of 4D-Var with Different Strategies for the Use of Adjoint Physics with the FSU Global Spectral Model , 2000 .

[92]  X. Zou,et al.  A Numerical Study of the Effect of GOES Sounder Cloud-Cleared Brightness Temperatures on the Prediction of Hurricane Felix , 2001 .

[93]  Olivier Talagrand,et al.  On extending the limits of variational assimilation in nonlinear chaotic systems , 1996 .

[94]  K. Kiwiel.  Methods of Descent for Nondifferentiable Optimization , 1985 .

[95]  Guanrong Chen,et al.  Kalman Filtering with Real-time Applications , 1987 .

[96]  Ionel M. Navon,et al.  Second-Order Information in Data Assimilation* , 2002 .

[97]  Ionel Michael Navon,et al.  Practical and theoretical aspects of adjoint parameter estimation and identifiability in meteorology and oceanography , 1998 .

[98]  L. Fillion.  Variational Assimilation of Precipitation Data and Gravity Wave Excitation , 2002 .

[99]  Virginie Marécal,et al.  Variational Retrieval of Temperature and Humidity Profiles from TRMM Precipitation Data , 2000 .

[100]  Moti Segal,et al.  Sensitivity of forecast rainfall in a Texas convective system to soil moisture and convective parameterization , 2000 .

[101]  Martin Ehrendorfer,et al.  Four-Dimensional Data Assimilation: Comparison of Variational and Sequential Algorithms , 1992 .

[102]  T. Vukicevic,et al.  The Effect of Linearization Errors on 4DVAR Data Assimilation , 1998 .

[103]  Ying-Hwa Kuo,et al.  Incorporating the SSM/I-Derived Precipitable Water and Rainfall Rate into a Numerical Model: A Case Study for the ERICA IOP-4 Cyclone , 2000 .

[104]  P. Courtier,et al.  A strategy for operational implementation of 4D‐Var, using an incremental approach , 1994 .

[105]  Heikki Järvinen,et al.  Variational assimilation of time sequences of surface observations with serially correlated errors , 1999 .

[106]  Philip E. Gill,et al.  Practical optimization , 1981 .

[107]  Thomas S. Pagano,et al.  Prelaunch characteristics of the Moderate Resolution Imaging Spectroradiometer (MODIS) on EOS-AM1 , 1998, IEEE Trans. Geosci. Remote. Sens..

[108]  W. Chao,et al.  Development of a four-dimensional variational analysis system using the adjoint method at GLA. I : dynamics , 1992 .

[109]  R. W. Lardner,et al.  Optimal control of open boundary conditions for a numerical tidal model , 1993 .

[110]  Juanzhen Sun,et al.  Dynamical and Microphysical Retrieval from Doppler Radar Observations Using a Cloud Model and Its Adjoint. Part I: Model Development and Simulated Data Experiments. , 1997 .

[111]  S. Cohn,et al.  Ooce Note Series on Global Modeling and Data Assimilation Construction of Correlation Functions in Two and Three Dimensions and Convolution Covariance Functions , 2022 .

[112]  T. Tsuyuki,et al.  Variational Data Assimilation in the Tropics Using Precipitation Data. Part III: Assimilation of SSM/I Precipitation Rates , 1997 .

[113]  G. Evensen.  Using the Extended Kalman Filter with a Multilayer Quasi-Geostrophic Ocean Model , 1992 .

[114]  Jean-Noël Thépaut,et al.  Simplified and Regular Physical Parameterizations for Incremental Four-Dimensional Variational Assimilation , 1999 .

[115]  Jean-Noël Thépaut,et al.  Impact of the Digital Filter as a Weak Constraint in the Preoperational 4DVAR Assimilation System of Météo-France , 2001 .

[116]  François-Xavier Le Dimet,et al.  Numerical Experience with Limited-Memory Quasi-Newton and Truncated Newton Methods , 1993, SIAM J. Optim..

[117]  Juanzhen Sun,et al.  Recovery of Three-Dimensional Wind and Temperature Fields from Simulated Single-Doppler Radar Data , 1991 .

[118]  J. R. Eyre,et al.  Variational Assimilation of Remotely-Sensed Observations of the Atmosphere , 1997 .

[119]  R. Anthes.  Advances in the Understanding and Prediction of Cyclone Development with Limited-Area Fine-Mesh Models , 1990 .

[120]  A. N. Tikhonov,et al.  Solutions of ill-posed problems , 1977 .

[121]  Juanzhen Sun,et al.  Wind and Thermodynamic Retrieval from Single-Doppler Measurements of a Gust Front Observed during Phoenix II , 1994 .

[122]  Arlindo da Silva,et al.  Data assimilation in the presence of forecast bias , 1998 .

[123]  G. Wahba,et al.  Adaptive Tuning of Numerical Weather Prediction Models: Simultaneous Estimation of Weighting, Smoothing, and Physical Parameters , 1998 .

[124]  Dusanka Zupanski,et al.  Four-Dimensional Variational Data Assimilation for the Blizzard of 2000 , 2002 .

[125]  Ronald M. Errico,et al.  An Adjoint Examination of a Nudging Method for Data Assimilation , 1997 .

[126]  P. Gauthier,et al.  A validation of the incremental formulation of 4D variational data assimilation in a nonlinear barotropic flow , 1998 .

[127]  Tadashi Tsuyuki,et al.  Variational data assimilation in the tropics using precipitation data part I: Column model , 1996 .

[128]  William Carlisle Thacker,et al.  The role of the Hessian matrix in fitting models to measurements , 1989 .

[129]  T. N. Krishnamurti,et al.  Physical initialization for numerical weather prediction over the tropics , 1991 .

[130]  R. S. Bell,et al.  The Meteorological Office analysis correction data assimilation scheme , 1991 .

[131]  Ionel M. Navon,et al.  Control Of Gravitational Oscillations in Variational Data Assimilation , 1993 .

[132]  R. Errico,et al.  Linearization and adjoint of parameterized moist diabatic processes , 1993 .

[133]  Numerical Simulation of Intense Precipitation Events South of the Alps: Sensitivity to Initial Conditions and Horizontal Resolution , 2000 .

[134]  J. Lions.  Optimal Control of Systems Governed by Partial Differential Equations , 1971 .

[135]  I. Michael Navon,et al.  The analysis of an ill-posed problem using multi-scale resolution and second-order adjoint techniques , 2001 .

[136]  Ionel M. Navon,et al.  Optimality of variational data assimilation and its relationship with the Kalman filter and smoother , 2001 .

[137]  W. Thacker,et al.  Fitting dynamics to data , 1988 .

[138]  Philippe Courtier,et al.  A New Hessian Preconditioning Method Applied to Variational Data Assimilation Experiments Using NASA General Circulation Models , 1996 .

[139]  Stanley Q. Kidder,et al.  Satellite Meteorology: An Introduction , 1995 .

[140]  P. Courtier,et al.  Four-dimensional variational data assimilation using the adjoint of a multilevel primitive-equation model , 1991 .

[141]  R. Treadon,et al.  A Tutorial on Lateral Boundary Conditions as a Basic and Potentially Serious Limitation to Regional Numerical Weather Prediction , 1997 .

[142]  P. Courtier,et al.  The ECMWF implementation of three‐dimensional variational assimilation (3D‐Var). II: Structure functions , 1998 .

[143]  Claude Lemaréchal,et al.  Some numerical experiments with variable-storage quasi-Newton algorithms , 1989, Math. Program..

[144]  Eugenia Kalnay,et al.  Atmospheric Modeling, Data Assimilation and Predictability , 2002 .

[145]  O. Axelsson.  Iterative solution methods , 1995 .

[146]  H. C. Davies,et al.  An Alpine Rainstorm: Sensitivity to the Mesoscale Upper-Level Structure , 2000 .

[147]  K. Droegemeier,et al.  The Advanced Regional Prediction System (ARPS) – A multi-scale nonhydrostatic atmospheric simulation and prediction model. Part I: Model dynamics and verification , 2000 .

[148]  Robert Atlas,et al.  Assimilation of SSM/I-Derived Surface Rainfall and Total Precipitable Water for Improving the GEOS Analysis for Climate Studies , 2000 .

[149]  Kelvin K. Droegemeier,et al.  Application of a New Adjoint Newton Algorithm to the 3D ARPS Storm-Scale Model Using Simulated Data , 1997 .

[150]  G. Evensen.  Sequential data assimilation with a nonlinear quasi‐geostrophic model using Monte Carlo methods to forecast error statistics , 1994 .

[151]  Ionel M. Navon,et al.  Incomplete observations and control of gravity waves in variational data assimilation , 1992 .

[152]  G. Browning,et al.  Discontinuous Forcing Generating Rough Initial Conditions in 4DVAR Data Assimilation , 2000 .

[153]  A. O'Neill.  Atmospheric Data Assimilation , 2000 .

[154]  William R. Cotton,et al.  Fitting Microphysical Observations of Nonsteady Convective Clouds to a Numerical Model: An Application of the Adjoint Technique of Data Assimilation to a Kinematic Model , 1993 .

[155]  Philippe Courtier,et al.  Interactions of Dynamics and Observations in a Four-Dimensional Variational Assimilation , 1993 .

[156]  R. W. Lardner,et al.  Optimal estimation of eddy viscosity for a quasi‐three‐dimensional numerical tidal and storm surge model , 1994 .

[157]  Zhi Wang,et al.  The second order adjoint analysis: Theory and applications , 1992 .

[158]  X. Zou,et al.  Initialization and Simulation of a Landfalling Hurricane Using a Variational Bogus Data Assimilation Scheme , 2000 .

[159]  Jean-François Mahfouf,et al.  Influence of physical processes on the tangent‐linear approximation , 1999 .

[160]  D. Zupanski,et al.  The effects of discontinuities in the Betts’Miller cumulus convection scheme on four-dimensional variational data assimilation , 1993 .

[161]  Yoshikazu Sasaki.  NUMERICAL VARIATIONAL ANALYSIS WITH WEAK CONSTRAINT AND APPLICATION TO SURFACE ANALYSIS OF SEVERE STORM GUST , 1970 .

[162]  V. M. Karyampudi,et al.  The Impact of Assimilating Satellite-Derived Precipitation Rates on Numerical Simulations of the ERICA IOP 4 Cyclone , 1994 .

[163]  Yoram J. Kaufman,et al.  Earth Observing System AM1 mission to Earth , 1998, IEEE Trans. Geosci. Remote. Sens..

[164]  John Derber,et al.  The National Meteorological Center's spectral-statistical interpolation analysis system , 1992 .

[165]  S. Cohn,et al.  Assessing the Effects of Data Selection with the DAO Physical-Space Statistical Analysis System* , 1998 .

[166]  R. Treadon,et al.  Physical initialization in the NMC Global Data Assimilation System , 1996 .

[167]  J. Derber,et al.  Variational Data Assimilation with an Adiabatic Version of the NMC Spectral Model , 1992 .

[168]  G. I. Marchuk,et al.  Formulation of the theory of perturbations for complicated models , 1975 .

[169]  R. Daley.  The Effect of Serially Correlated Observation and Model Error on Atmospheric Data Assimilation , 1992 .

[170]  Peter M. Lyster,et al.  Assimilation of Stratospheric Chemical Tracer Observations Using a Kalman Filter. Part I: Formulation , 2000 .

[171]  P. Houtekamer,et al.  Data Assimilation Using an Ensemble Kalman Filter Technique , 1998 .

[172]  J. Nocedal.  Updating Quasi-Newton Matrices With Limited Storage , 1980 .

[173]  Jacques Verron,et al.  Sensitivity Analysis in Variational Data Assimilation (gtSpecial IssueltData Assimilation in Meteology and Oceanography: Theory and Practice) , 1997 .

[174]  P. Courtier,et al.  Variational Data Assimilation with a Semi-Lagrangian Semi-implicit Global Shallow-Water Equation Model and Its Adjoint , 1993 .

[175]  Lisan Yu,et al.  Variational Estimation of the Wind Stress Drag Coefficient and the Oceanic Eddy Viscosity Profile , 1991 .

[176]  X. Zou,et al.  Rainfall Assimilation through an Optimal Control of Initial and Boundary Conditions in a Limited-Area Mesoscale Model , 1996 .

[177]  An initialization method to incorporate precipitation data into a mesoscale numerical weather prediction model , 1993 .

[178]  Jeffrey L. Anderson.  An Ensemble Adjustment Kalman Filter for Data Assimilation , 2001 .

[179]  Jorge Nocedal,et al.  On the limited memory BFGS method for large scale optimization , 1989, Math. Program..

[180]  Dick Dee,et al.  Observability of discretized partial differential equations , 1988 .

[181]  D. Zupanski.  A General Weak Constraint Applicable to Operational 4DVAR Data Assimilation Systems , 1997 .