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Tellus A: Dynamic Meteorology and Oceanography Volume 45, 1993 - Issue 5
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Original Articles

Important literature on the use of adjoint, variational methods and the Kalman filter in meteorology

Philippe CourtierECMWF, Shinfield Park, Reading RG2 9AX, UK;
,
John DerberNMC, Washington, DC 20233, USA;
,
Ron ErricoNCAR, Boulder, Colorado 80307, USA;
,
Jean-Francois LouisAER, 840 Memorial Drive, Cambridge, Massachusetts02139, USA
&
Tomislava VukiĆEviĆNCAR, Boulder, Colorado 80307, USA;
Pages 342-357 | Received 27 Apr 1993, Accepted 24 Jun 1993, Published online: 15 Dec 2016

References

  • Sasaki, Y. 1955 (V). A fundamental study of the numerical prediction based on the variational principle. J. Meteor. Soc. Japan 33, 262–275.
  • —Applies Hamilton principle to the derivation of the equation of motion.
  • Sasaki, Y. 1958 (V). An objective analysis based on the variational method. J. Meteor. Soc. Japan 36, 77–88.
  • —Applies variational methods to find balanced fields.
  • Sasaki, Y. 1960 (V). An objective analysis for determining initial conditions for the primitive equations. Tech. Rep. 208. Department of Oceanography and Meteorology, A & M College of Texas.
  • Washington, W. M. and R. T. Duquet, 1963 (V). An objective analysis of stratospheric data by Sasaki's method. Department of Meteorology, Penn. State Univ., 23 pp.
  • Jones, R. H. 1965 (K). Optimal estimation of initial conditions for numerical prediction. J. Atmos. Sci. 22, 658–663.
  • — Introduces the idea of Kalman filtering in meteorology.
  • Lorenz, E. N. 1965 (I). A study of the predictability of a 28-variable atmosphere model. Tellus 17, 321–333.
  • Stephens, J. J. 1965 (V). A variational approach to numerical weather prediction. Rep. No. 3, Atmos. Sci. Group. The University of Texas, Austin, USA, 243 pp.
  • —Stephen's PhD dissertation, in which Sasaki's ideas were applied.
  • Gruber, A. and J. J. O'Brien, 1968 (V). An objective analysis of wind data for energy budget studies. J. App!. Meteor. 7, 333–338.
  • —This paper discusses how to find a non divergent analysis using Lagrange multipliers. The authors were probably not aware of Sasaki's work at this time.
  • Petersen, D. P. 1968 (K). On the concept and implementation of sequential analysis for linear random fields. Tellus 20, 673–686.
  • — Proposes an implementation of Kalman filter relying upon Green's functions.
  • Epstein, E. S. 1969 (K). Stochastic dynamic prediction. Tellus 21, 739–759.
  • —Implements what could be considered as a Kalman filter. The author was apparently unaware of Kalman's work.
  • Sasaki, Y. 1969 (V). Proposed inclusion of time variation terms, observational and theoretical, in numerical variational objective analysis. J. Meteor. Soc. Japan 47, 115–124.
  • —Uses linear advection and diffusion as a constraint.
  • Thompson, P. D. 1969 (V). Reduction of analysis error through constraints of dynamical consistency. J. AppL Meteor. 8, 738–742.
  • —Fits analyses to dynamics using a linearized vorticity equation. Theoretical paper. Expresses the need for appropriate numerical methods for more realistic models.
  • Jazwinski, A. H. 1970 (K). Stochastic processes and filtering theory. Academic press, New York, 376 pp.
  • —Basic book on Kalman filtering. Demonstrate the (already known) equivalence Kalman filtering/4D-Var: A linear regression is equivalent to a least-square fit.
  • Flattery, T. W. 1970 (V). Spectral analysis and forecasting. Numerical Weather prediction activities, National Meteorological Center, Second half 1969,51 pp.
  • —Uses Hough modes for objective analysis. Ancestor of NMC's SSI and Meteo-France and ECMWF's 3D-Var.
  • O'Brien, J. J. 1970 (V). Alternative solutions to the classical vertical velocity problem. J. App!. Meteor. 9, 197–203.
  • — Applies Sasaki's formalism for solving the omega equation.
  • Sasaki, Y. 1970 (V). Some basic formalisms in numerical variational analysis. Mon. Wea. Rev. 98, 875–883.
  • —Basic paper which introduces the variational approach with the weak and strong constraints concepts. Ahead of its time in that the numerical tools were lacking.
  • Sasaki, Y. 1970 (V). Numerical variational analysis formulated under the constraints as determined by longwave equations and a low pass filter. Mon. Wea. Rev. 98, 884–898.
  • —Applies variational formalism to filter high frequency oscillations in a shallow-water model.
  • Sasaki, Y. 1970 (V). Numerical variational analysis with weak constraint and application to surface analysis of severe storm gust. Mon. Wea. Rev. 98, 899–910.
  • —Uses two-dimensional linear advection as a weak constraint.
  • Sasaki, Y. and J. Lewis, 1970 (V). Numerical variational objective analysis of the planetary boundary layer in conjunction with squall line formation. J. Meteor. Soc. Japan 48, 381–398.
  • —Practical application of Sasaki's formalism.
  • Stephens, J. J. 1970 (V). Variational initialization with the balance equation. J. AppL Meteor. 9, 732–739.
  • —Applies Sasaki's formulation to a 2-D initialization problem.
  • Lions, J. L. 1971 (G). Optimal control of systems governed by partial differential equations (English translation). Springer-Verlag, Berlin.
  • —General introduction to the application of functional analysis to optimal control.
  • Sasaki, Y. 1971 (V). A theoretical interpretation of anisotropically weighted smoothing on the basis of numerical variational analysis. Mon. Wea. Rev. 99, 698–708.
  • —Uses Green's function to justify some enhancements of Cressman iterative analysis scheme.
  • Wagner, K. 1971 (V). Variational analysis using observational and low pass filtering constraints. M.S. thesis. Univ. of Oklahoma, Norman, 39 pp.
  • Lewis, J. M. 1972 (V). An operational upper air analysis. Tellus 24, 514–530.
  • —Uses Sasaki's formalism to obtain an analysis which projects on the slowly evolving solutions of a model.
  • Lewis, J. M. and T. H. Grayson, 1972 (V). The adjustment of surface wind and pressure by Sasaki's variational matching technique. J. App!. Meteor. 11, 586–597.
  • —Find slowly varying (balanced) fields using a 2-D model.
  • Petersen, D. P. 1973 (K). Transient suppression in optimal sequential analysis. J. Applied Meteor. 12, 437–440.
  • —Proves that Kalman filter preserves geostrophy. Remarks that any assimilation scheme will have to be adaptive.
  • Petersen, D. P. 1973 (K). A comparison of the performance of quasi-optimal and conventional objective analysis schemes. J. Applied Meteor. 12, 1093–1101.
  • —Attempts at a proper comparison of operational and statistical analysis optimal schemes.
  • Sheets, R. C. 1973 (V). Analysis of hurricane data using the variational optimization approach with a dynamic constraint. J. AppL Meteor. 9, 732–739.
  • Marchuk, G. I. 1974 (G((Russian version 1967). Numerical methods in weather prediction. Academic Press, New-York, 277 pp.
  • —Chapter 5 is dedicated to the use of the adjoint formalism to derive integral conserving numerical schemes.
  • Marchuk, G. I.. 1974 (SG((Russian version 1967). The numerical solution of problems of atmospheric and oceanic dynamics, 387 pp., Gidrometeoizdat, Leningrad, USSR. (English translation, Rainbow Systems, Alexandria, va.).
  • —Presents some application in meteorology of the adjoint formalism.
  • Achtemeier, G. C. 1975 (V). On the initialization problem: a variational adjustment method. Mon. Wea. Rev. 103, 1089–1103.
  • —Applies Sasaki's formalism to the initialization problem.
  • Marchuk, G. I. 1975 (VG). Methods of numerical mathematics. Springer-Verlag, 313 pp.
  • —Chapter 5 applies adjoint technique to inverse problems.
  • Marchuk, G. I. and Y. K. Skriba, 1976 (S). Numerical calculation of the conjugate problem for a model of the thermal interaction of the atmosphere with the oceans and continents. Izvestiya Atm. and Ocean Phys. 12, 5,459–469.
  • —Use adjoint equations to compute sensitivity of climate anomalies to surface temperature.
  • Penenko, V. V. and N. N. Obratsov, 1976 (A). A variational initialization method for the fields of the meteorological elements (English translation). Soviet Meteorology and Hydrology 11, 1–11.
  • —Apply 4D-Var assimilation to a linear primitive equation model minimizing a cost function. The gradient is computed using a discretized form of the continuous adjoint equations. Uses analyses as “observations”.
  • Sasaki, Y. 1976 (G). Variational design of finite difference schemes for initial value problems with an integral invariant. J. Comp. Phys. 21, 270–278.
  • —Similar idea as Marchuk (1974) although it uses the Euler—Lagrange equations.
  • Gordin, V. A. 1977 (V). Variational adjustment of meteorological fields. Soviet Meteorology and Hydrology 12, 95–96.
  • —Uses variational framework for imposing geostrophy. Mentions application to time dependent problems.
  • McFarland, M. S. and Y. K. Sasaki, 1977 (V). Variational analysis of temperature and moisture advection in a severe storm environment. J. Met. Japan 55, 421–430.
  • Marchuk, G. I. and G. P. Kurbatkin, 1977 (S). Physical and mathematical aspects of weather analysis and forecasting. Soviet Meteorology and Hydrology 11, 25–33.
  • —Explains how the adjoint can be used for sensitivity experiments. Presents results for climate.
  • Penenko, V. V. 1977 (PG). A numerical model of the thermal regime of the atmosphere. Izvestiya Atm. and Ocean Phys. 13, 6,401–407.
  • —Proposes to use the adjoint in order to compute gradients of a functional for a parameter estimation.
  • Penenko, V. V. 1977 (G). Energy balanced discrete models of atmospheric process dynamics. (English translation) Soviet Meteorology and Hydrology 10, 3–20.
  • —Uses adjoint formalism to derive energy conserving finite difference schemes.
  • Sadokov, V. D. and D. B. Steinbok, 1977 (S). Application of conjugate functions in analysis and fore-casting of the temperature anomaly. (English translation) Soviet Meteorology and Hydrology 10, 16–21.
  • —Computes sensitivity of 30-day average tropospheric temperature anomaly to the ocean surface temperature using the adjoint of a simple model.
  • Seaman, R. S., R. Falconer and J. Brown, 1977 (V). Application of a variational blending technique to numerical analysis in the Australian region. Austr. Meteor. Mag. 3-22.
  • Daley, R. 1978 (V). Variational non-linear normal mode initialization. Tel/us 30, 201–218.
  • —Variational assimilation with a Machenhauer balance constraint (tendencies of gravity waves approximately set to zero).
  • Lewis, J. M. and S. C. Bloom, 1978 (V). Incorporation of time continuity into subsynoptic analysis by using dynamical constraints. Tel/us 30, 496–516.
  • —Surface wind and pressure analyses are coupled in time using a simple horizontal momentum equation model.
  • Stephens, J. J. and K. W. Johnson, 1978 (V). Middle-large-scale variational adjustment of atmospheric fields in mesoscale diagnostic numerical variational analysis models. Final report, Department of Meteorology, Florida State Univ., 1–38.
  • Marchuk, G. I. and V. V. Penenko, 1979 (S). A study of the sensitivity of discrete models of atmospheric and oceanic dynamics. Izvestiya Atm. and Ocean Phys. 15, 11, 785–789.
  • —Presents how the adjoint can be used for sensitivity application. Discusses the importance of having the adjoint of the discrete model.
  • Ghil, M., S. Cohn, J. Tvantzis, K. Bube and E. Isaacson, 1980 (K). Application of estimation theory to numerical weather prediction. 1980 ECMWF seminar on “Data assimilation methods,” pp. 249-334. Also in Dynamic Meteorology, Data assimilation methods. (L. Bengtsson, M. Ghil, E. Källen, editors), Springer-Verlag, 1981.
  • —Popularises Kalman filtering in Meteorology. Theory and applications to linear shallow-water models.
  • Lewis, J. M.. 1980 (V). Dynamical adjustment of 500 mb vorticity using P. D. Thompson's scheme: a case study. Tellus 32, 511–524.
  • —Uses a linearised vorticity equation to fit analyses.
  • Wahba, G. and J. Wendelberger, 1980 (V). Some new mathematical methods for variational analysis using splines and cross-validation. Mon. Wea. Rev. 108, 1122–1143.
  • —Integrates variational formalism and statistical formulation. Authors stress that the raw data should be used directly with no prior gridding and that the radiances should be used directly with no prior inversion. The importance of the paper has been underestimated in the 1980s, mainly because General Cross Validation is more appropriate for case studies than for operational problems where statistics are available from the past.
  • Cacucci, D. G. 1981 (S). Sensitivity theory for nonlinear systems. I. Nonlinear functional analysis approach. J. Math. Phys. 22, 2794–2802.
  • —Theoretical introduction to the adjoint formalism applied to sensitivity experiments.
  • Navon, I. M. 1981 (V). Implementation of “a posteriori” methods for enforcing conservation of potential enstrophy and mass in discretized shallow-water equations models. Mon. Wea. Rev. 109, 946–958.
  • Phillips, N. A. 1981 (V). Variational analysis and the slow manifold. Mon. Wea. Rev. 109, 2415–2426.
  • Sasaki, Y. K. and J. McGinley, 1981 (V). Application of the inequality constraints in adjustments of superadiabatic layers. Mon. Wea. Rev. 110, 1635–1644.
  • Cohn, S. E. 1982 (K). Methods of sequential estimation for determining initial data in numerical weather prediction. PhD Thesis. Published as report CI-6-82, Courant Institute, New York University, 183 pp.
  • Ghil, M., S. E. Cohn and A. Dalcher, 1982 (K). Sequential estimation, data assimilation and initializa-tion. In: The Interaction Between Objective Analysis and Initialization (ed., D. Williamson), Publ. MeteoroL 127, (Proc. 14th Stanstead Seminar), McGill Univ., Montreal, 83–97.
  • Hall, M. C. G., D. G. Cacucci and M. E. Schlesinger, 1982 (S). Sensitivity Analysis of a Radiative-Convective Model by the Adjoint Method. J. Atmos. Sci. 39, 2038–2050.
  • —Use the 2 layer Oregon State University model physics. Integrates a radiative-convective model until equilibrium. Compute the sensitivity of asymptotic temperature to 312 parameters (including basic physical constants).
  • Hoffman, R. N. 1982 (V). SASS wind ambiguity removal by direct minimization. Mon. Wea. Rev. 110, 434–445.
  • Le Dimet, F.-X. 1982 (G). A general formalism of variational analysis. CIMMS report 22, Cooperative Institute for Mesoscale Meteorological studies, 815 Jenkins Street, Norman, Ok., 73019, 34 pp.
  • —Proposes several algorithms for solving variational problems including the use of adjoint.
  • Lewis, J. M. 1982 (V). Adaptation of Thompson's scheme to the constraint of potential vorticity conservation. Mon. Wea. Rev. 110, 1618–1634.
  • —Fits 2-layer quasigeostrophic baroclinic model to analyses.
  • Marchuk, G. I. 1982 (S). Mathematical issues of industrial effluent optimization. J. Meteor. Soc. Japan 60, 481–485.
  • Navon, I. M. 1982 (V). A posteriori numerical techniques for enforcing simultaneous conservation of integral invariants upon finite-difference shallow-water equations models. Notes on Numerical fluid dynamics, vol. 5. (Henri-Viviand Vieweg ed.) 230–240.
  • Tribbia, J. J. 1982 (V). On variational normal mode initialization. Mon. Wea. Rev. 110, 455–470.
  • —A detailed analysis of Daley's approach and some extensions, with applications to a shallow water model.
  • Williamson, D. 1982 (VK). The interaction between objective analysis and initialization. Publ. Meteorol. 127 (Proc. 14th Stanstead seminar), Mc Gill Univ., Montreal.
  • —Several papers on variational initialization and on Kalman filter.
  • Balgovind, R., A. Dalcher, M. Ghil and E. Kalnay, 1983 (K). A stochastic-dynamic model for the spatial structure of forecast error statistics. Mon. Wea. Rev. 111,701–722.
  • Bloom, S. C. 1983 (V). The use of dynamical constraints in the analysis of mesoscale rawind sonde data. Tel/us 35A, 363–378.
  • —Finds an analysed state which approximately satisfies a 2-D evolution equation.
  • Hall, M. C. G. and D. G. Cacucci, 1983 (S). Physical interpretation of the adjoint functions for sensitivity analysis of atmospheric models. J. Atmos. Sci. 40, 2537–2546.
  • —Shows that the adjoint solution can be used as a whole for sensitivity experiments.
  • Lewis, J. M. and L. Panetta, 1983 (V). The extension of P. D. Thompson's scheme to multiple time levels. J. Clim. App!. Meteorol. 22,1649–1653.
  • —Generalize their previous work to several time levels.
  • Navon, I. M. and R. de Villiers, 1983 (GO). Combined penalty multiplier optimization methods to enforce integral invariants conservation. Mon. Wea. Rev. 111,1228–1243.
  • Puri, K. 1983 (V). Some experiments in variational normal mode initialization in data assimilation. Mon. Wea. Rev. 111,1208–1218.
  • Cacucci, D. G. and M. C. G. Hall, 1984 (S). Efficient estimation of feedback effects with application to climate models. J. Atmos. Sci. 41,2063–2068.
  • —Proposes the use of the adjoint formalism for the estimation of sensitivity to feedback loops.
  • Courtier, P. 1984 (A). Presentation d'une méthode variationnelle quadridimensionelle d'assimilation de données distribuées dans l'espace et dans le temps. Note de travail de l'EERM no. 101. Available from Météo-France, Paris.
  • —Results obtained with the barotropic vorticity equation and real wind and height radiosondes observations. Far better described in Courtier and Talagrand (1987).
  • Hoffman, R. N. 1984 (V). SASS wind ambiguity removal by direct minimization. Part II: use of smoothness and dynamical constraints. Mon. Wea. Rev. 112,1829–1852.
  • Temperton, C. 1984 (V). Variational normal mode initialization for a multilevel model. Mon. Wea. Rev. 112, 2303–2316.
  • Courtier, P. 1985 (A). Experiments in data assimilation using the adjoint model technique. Workshop on High-resolution analysis, ECMWF, UK, June 1985,20 pp.
  • —In addition to the vorticity equation results, this paper demonstrates how to control gravity waves in 4D-Variational Assimilation. Better described in Courtier and Talagrand (1990).
  • Derber, J. C. 1985 (A). The variational four-dimensional assimilation of analyses using filtered models as constraints. PhD Thesis, University of Wisconsin—Madison, 142 pp.
  • Lewis, J. M. and J. C. Derber, 1985 (A). The use of adjoint equations to solve a variational adjustment problem with advective constraints. Tel/us 37A, 309–322.
  • — First, test the method on the 1D advection equation, then real data test using the quasi-geostrophic potential vorticity equation and analyses which were produced using Barnes scheme applied to RAOBS and VAS data.
  • Marchuk, G. I., Y. K. Skriba and I. G. Protsenko, 1985 (K). Method of calculating the evolution of ran-dom hydrodynamic fields on the basis of adjoint equations. Izvestya Atm. and Ocean Phys. 21, 2, 87–92.
  • —Propose to use the fact that the adjoint equations lead to local influence to simplify the transport in time of the covariance matrix of forecast errors.
  • Marchuk, G. I., Y. K. Skriba and I. G. Protsenko, 1985 (K). Application of adjoint equations to problem of estimating the state of random hydrodynamic fields. Izvestya Atm. and Ocean Phys. 21, 3, 175–184.
  • —Implement a Kalman filter based on their simplification of the transport in time of the covariance matrix of forecast errors.
  • Parrish, D. F. and S. F. Cohn, 1985 (K). A Kalman filter for a two-dimensional shallow-water model: formulation and preliminary experiments. NMC office note no. 304.
  • Talagrand, O. and P. Courtier, 1985 (GPA). Formalisation de la methode du modele adjoint. Applications meteorologiques. Note de Travail No. 117. EERM, Paris, France. 10 pp.
  • —Theoretical introduction to adjoint model. Describes adjoint with respect both to initial conditions and to parameters.
  • Urban, B. 1985 (I). Maximal error amplification in simple meteorological models (in French). Working note, Ecole Nationale de la Metéorologie, Toulouse, France.
  • — Finds most unstable modes in the energetic sense of vorticity equation model and 2-layer Q.G. model.
  • Budgell, N. P.. 1986 (K). Nonlinear data assimilation for shallow-water equations in branched channels. J. Geophys. Res. 91, 10633–10644.
  • Courtier, P. 1986 (G). Determination pratique de l'adjoint d'un modele de prevision. Note de Travail No. 159. EERM, Paris, France.
  • —Describes the way the author derived the adjoint of a vorticity equation model and a shallow-water equation model (not to be followed any longer). Presents two ways of validating the adjoint code which should be systematically followed. (Green identity and Taylor formula).
  • Courtier, P. 1986 (SP). Le modele adjoint, outil pour des experiences de sensibilite. Note de Travail No. 166. EERM, Paris, France. 36 pp.
  • —Use of primitive equations for a 2D fluid on a sphere (shallow-water equation). Looks at the sensitivity to initial conditions, to the initialization process and to orography. Uses it to diagnose tidal waves. Example of misuse: find the orography that minimizes the forecast error.
  • Hoffman, R. N. 1986 (V). A four-dimensional analysis exactly satisfying equations of motion. Mon. Wea. Rev. 114, 388–397.
  • —An example of 4D-variational assimilation without the use of adjoint equations.
  • Le Dimet, F. X. and O. Talagrand, 1986 (G). Variational algorithms for analysis and assimilation of meteorological observations: theoretical aspects. Tellus 38A, 97–100.
  • —Describes several variational algorithms introducing the “augmented Lagrangian” and the adjoint method for data assimilation.
  • Lorenc, A. C. 1986 (V). Analysis methods for numerical weather prediction. Q. J. R. Meteorol. Soc. 112, 1177–1194.
  • Sasaki, Y. K. 1986 (VAG). Variational methods in geosciences. Proceedings of the International Sym-posium on Variational methods in geosciences, University of Oklahoma. Norman, OK., 15-17 October 1985. Elsevier, 309 pp.
  • —In addition to papers published later in open literature, we should mention a good review paper by I. M. Navon on variational and optimization methods in meteorology and an introduction to the application of optimal control to meteorological problems by O. Talagrand.
  • Talagrand, O. 1986 (G). Application of optimal control to meteorological problems. In: Variational methods in geosciences, vol. 5. ([Y. K. Sasaki, ed.) Elsevier, pp. 13–28.
  • —Mathematical description. Short overview of meteorological applications.
  • Courtier, P. 1987 (A). Assimilation variationnelle d'observations météorologiques a l'aide des equations de Saint- Venant directes et adjointes. Note de Travail No. 175. EERM, Paris, France. 62 pp.
  • —Presents the control of the gravity waves in variational assimilation of real observations relying on the shallow-water equations. Results also described (more briefly) in Courtier and Talagrand, 1990.
  • Courtier, P. 1987 (ASPG). Application du contrôle optimal it la prévision numérique en méteorologie. These de Doctorat, Université Paris, France.
  • —Gathers all previous publications on this subject. The adjoint of the 3-dimensional primitive equations is derived in continuous form.
  • Courtier, P. and O. Talagrand, 1987 (A). Variational assimilation of meteorological observations with the adjoint vorticity equation. Part II. Numerical results. Q. J. R. Meteorol. Soc. 113, 1329–1368.
  • —Uses the vorticity equation and all the data for a 24 hour period. Starts from an atmosphere at rest. Finds that the Aleutian low is reconstructed even though there were no observations there. Reduction of small scale noise by adding a smoothing term in the cost function.
  • Derber, J. C. 1987 (A). Variational four-dimensional analysis using quasi-geostrophic constraints. Mon. Wea. Rev. 115, 998–1008.
  • —Extension of the work by Lewis and Derber. Further tests of technique. With regular insertion, best fit is in the center of the data period. Used the same quasi-geostrophic model as in the previous work and either primitive equation simulation or satellite observations for data.
  • Navon, I. M. and D. M. Legler, 1987 (0). Conjugate-Gradient Methods for Large-Scale Minimization in Meteorology. Mon. Wea. Rev. 115, 1479–1502.
  • —Review of conjugate gradient methods; applies four methods to two problems.
  • Panchang, V. G. and J. J. O'Brien, 1987 (P). On the determination of hydraulic model parameters using the adjoint state formulation. In: Modelling marine systems, vol. 1. (A. M. Davies, ed.). CRC Press, Inc., Boca Raton, FL., USA, 18 pp.
  • —Estimates various parameters involved in modelling hydraulic systems, particularly the friction factor of tidal rivers. Uses the Lagrange multipliers form of the adjoint.
  • Talagrand, O. and P. Courtier, 1987 (GA). Variational assimilation of meteorological observations with the adjoint vorticity equation. Part 1: Theory. Q. J. R. MeteoroL Soc. 113, 1311–1328.
  • —Develops theory of adjoint assimilation from first principles. Applies to data assimilation for Haurwitz waves. Appendices give description of adjoint formalism in general, time stepping, handling of FFTs.
  • Farrell, B. 1988 (I). Optimal excitation of neutral Rossby waves. J. Atmos. Sci. 45, 163–172.
  • —One of the first papers on using adjoints for determination of optimal modes.
  • Lacara, J.-F. and O. Talagrand, 1988 (VI). Short-range evolution of small perturbations in a barotropic model. Tellus 40A, 81–95.
  • LeDimet, F. X. 1988. Determination of the adjoint of a numerical weather prediction model. Tech. Rep. FSU-SCR1-88-79, Florida State University, Talahassee, Florida, 32306-4052,22 pp.
  • Lorenc, A. C. 1988 (V). A Practical Approximation to Optimal Four-Dimensional Objective Analysis. Mon. Wea. Rev. 116, 730–745.
  • —Based on RAFS (Regional model operational at NMC, Washington). Linearized almost adiabatic backward forecast used for adjoint. The method did not improve forecasts, but this could be explained by several approximations introduced in the implementation.
  • Thacker, W. C. and R. Long, 1988. Fitting dynamics to data. J. Geophys. Res. 93, 10655–10665.
  • Vautard, R. and B. Legras, 1988 (PI). On the source of mid-latitude low frequency variability. Part II: Non linear equilibration of weather regimes. J. Atmos. ScL 45, 2845–2867.
  • —Shows that it is necessary to be very careful in using the adjoint technique for climatic application. They are successful using an ensemble average.
  • Derber, J. C. 1989 (AP). A variational continuous assimilation technique. Mon. Wea. Rev. 117, 2437–2446.
  • —The control parameter is a constant term in each dynamical equation. Uses fixed time weighting, which in most of the experiments is constant. Use in diagnosing model errors, which may be used in subse-quent forecasts. The method is numerically similar to the adjoint analysis, but yields superior forecasts in this simple test. Uses a quasi-geostrophic model. The gradient of the objective function is a weighted sum of the adjoint solution at each observation time.
  • Farrel, B. F. 1989 (I). Optimal excitation of baroclinic waves. J. Atmos. Sci. 46, 1193–1206.
  • Fillion, L. and C. Temperton, 1989 (V). Variational implicit normal mode initialization. Mon. Wea. Rev. 117, 2219–2229.
  • —Simplification and more efficient application of the proposals by Daley , (1978) and Tribbia (1983) obtained by solving the problem in physical rather than model space.
  • Andretta, A., V. Cuomo, R. Rizzi and C. Serio, 1990 (A). Cloud clearing schemes for atmospheric tem-perature soundings. Meteorology and environmental sciences (R. Guzzi, A. Navarra and J. Shukla, eds.). World Scientific Publishing Co., Singapore, 537–564.
  • —Compares two approaches, one of which uses a Kalman filter.
  • Bernardet, P. and L. Amodei, 1990 (A). Problems of adjoint methods in limited area modelling. Proceedings, World Meteorological Organisation, International Symposium on Assimilation of observations in meteorology and oceanography, Clermont-Ferrand, France, 9-13 July 1990, 109-113. Available from WMO.
  • —Discusses lateral boundary conditions using a 1-D advective equation.
  • Casse, V., A. Ratier and H. Roquet, 1990 (A). Direct variational assimilation of scatterometer back-scatter measurements into numerical weather prediction models. Proceedings, World Meteorological Organisation, International Symposium on Assimilation of observations in meteorology and oceanography, Clermont-Ferrand, France, 9-13 July 1990, 153-158.
  • —Progress towards direct use of TOVS radiances is reviewed and some remaining problems are discussed.
  • Courtier, P. and O. Talagrand, 1990 (A). Variational assimilation of meteorological observations with the direct and adjoint shallow-water equations. Tellus 42A, 531–549.
  • Courtier, P., J.-N. Thépaut and O. Talagrand, 1990 (A). 4-dimensional data assimilation using the adjoint of a primitive equation model. WMO. International Symposium on Assimilation of observations in meteorology and oceanography, Clermont-Ferrand (France), pp. 337–340.
  • —Early results with adjoint of a 3D PE model.
  • Douady, D. and O. Talagrand, 1990 (A). The impact of threshold processes on variational assimilation. Proceedings, WMO. International Symposium on Assimilation of observations in meteorology and oceanography, Clermont-Ferrand (France), 486–487. Available from WMO.
  • Eyre, J. R. 1990. Progress on direct use of satellite sounding radiances in numerical weather prediction. Proceedings, World Meteorological Organisation, International Symposium on Assimilation of observations in meteorology and oceanography, Clermont-Ferrand, France, 9-13 July 1990, 117-121. Available from WMO.
  • Farrel, B. F. 1990 (I). Small error dynamics and the predictability of atmospheric flows. J. Atmos. Sci. 47, 2409–2416.
  • Fillion, L. 1990 (V). Data assimilation using variational implicity normal mode initialization. Proceedings, World Meteorological Organisation, International Symposium on Assimilation of observations in meteorology and oceanography, Clermont-Ferrand, France, 9-13 July 1990, 591-596. Available from WMO.
  • —Combining 3-D analysis and initialization in one variational formalism. By applying the normal-mode balance constraint in physical rather than Hough-mode space, the variational weights may be defined locally and therefore more realistically.
  • Fredericksen, J. S. and R. C. Bell, 1990 (I). North Atlantic blocking during January 1979: linear theory. Q. J. R. Meteorol. Soc. 116, 1289–1313.
  • —Primarily a determination of exponential modes for realistic basic states, but also examines some optimal modes (based on initial tendencies only).
  • Ghil, M. 1990 (K). Sequential estimation in meteorology and oceanography: theory and numerics. Proceedings, World Meteorological Organisation, International Symposium on Assimilation of observations in meteorology and oceanography, Clermont-Ferrand, France, 9-13 July 1990, 85–90.
  • —Discussion of the theoretical framework for 4DDA.
  • Gollvik, S. and N. Gustafsson, 1990 (A). Data assimilation with adjoint shallow water models and application to mesoscale wind simulation. Proceedings, World Meteorological Organisation, International Symposium on Assimilation of observations in meteorology and oceanography, Clermont-Ferrand, France, 9-13 July 1990, 31-36. Available from WMO.
  • —Investigation of boundary layer assimilation using a reduced-gravity shallow water model.
  • Kapitza, H., Y. Li and K. Droegemeier, 1990 (A). Sensitivity experiments with the adjoint of a non-hydrostatic mesoscale model. Proceedings, World Meteorological Organisation, International Symposium on Assimilation of observations in meteorology and oceanography, Clermont-Ferrand, France, 9-13 July 1990, 25-30. Available from WMO.
  • —Data assimilation using Doppler radar data is also discussed.
  • Kiselnikova, V. Z. 1990. The use of satellite and radar data in the mesoscale analysis of atmospheric moisture. Proceedings, World Meteorological Organisation, International Symposium on Assimilation of observations in meteorology and oceanography, Clermont-Ferrand, France, 9-13 July 1990, 171-173. Available from WMO.
  • Louis, J.-F., R. N. Hoffman and R. G. Isaacs, 1990 (V). Improved use of satellite data in the GL meteorological analysis system. Proceedings, World Meteorological Organisation, International Symposium on Assimilation of observations in meteorology and oceanography, Clermont-Ferrand, France, 9-13 July 1990, 136-140. Available from WMO.— General review.
  • Louis, J.-F., R. N. Hoffman and T. Nehrkorn, 1990 (E). Optimizing a local weather forecast model. NSF Contract No. ISI-8960592, Final Report. 32 pp. Available from AER, Inc., 840 Memorial Drive, Cambridge, Mass 02139, USA.
  • Mahfouf, J.-F. 1990 (V). A variational assimilation of soil moisture in meteorological models. Proceedings, World Meteorological Organisation, International Symposium on Assimilation of observations in meteorology and oceanography, Clermont-Ferrand, France, 9-13 July 1990, 249-254. Available from WMO.
  • Marchuk, G. I. and Y. K. Skriba, 1990 (S). Role of the adjoint functions in studying the sensitivity of a model of the thermal interaction of the atmosphere and ocean to variation in input data. Izvestiya Atm. and Ocean Phys. 26, 5, 335–342.
  • —Uses adjoint equations to compute sensitivity of climate anomalies to ocean conditions.
  • Navon, I. M., X. Zou, K. Johnson, J. Derber and J. Sela, 1990 (V). Variational Real-Data Assimilation with the NMC Spectral Model. WMO. International Symposium on Assimilation of observations in meteorology and oceanography, Clermont-Ferrand, France, pp. 341-348. Available from WMO.
  • Pailleux, J. 1990. A global variational assimilation scheme and its application for using TOVS radiances. Proceedings, World Meteorological Organisation, International Symposium on Assimilation of observations in meteorology and oceanography, Clermont-Ferrand, France, 9-13 July 1990, 325-328. Presentation of 3D-Variational Assimilation plans. Available from WMO.
  • —Introduces the 3D-Variational Assimilation project of the European Centre for Medium-Range Weather Forecasts.
  • Ramamurthy, M. K. and I. M. Navon, 1990 (G). Application of a conjugate-gradient method to varia-tional assimilation of meteorological fields. Proceedings, World Meteorological Organisation, International Symposium on Assimilation of observations in meteorology and oceanography, Clermont-Ferrand, France, 9-13 July 1990, 359-364. Available from WMO.
  • Thepaut, J.-N. and P. Moll, 1990 (A). Variational inversion of simulated TOYS radiances using the adjoint technique. Q. J. R. MeteoroL Soc. 116, 1425–1448.
  • —Uses exact linear tangent operator for TOYS export package. Show that the nonlinearity, at least for midlatitudes, is weak except for HIRS channels 11 and 12 which are quite sensitive to water vapor. Eigenvector analysis of how much information comes from first guess observations (6 pieces of info for the 19 HIRS channels, 3 pieces for the 4 MSU channels, but only 7 for the two instruments combined). Variational retrieval tested in simulation. Quasi-Newton technique used. Five to ten iterations required depending on conditioning.
  • Todling, R. and M. Ghil, 1990 (K). Kalman filtering for a two-layer, two-dimensional shallow-water model. Proceedings, World Meteorological Organisation, International Symposium on Assimilation of observations in meteorology and oceanography, Clermont-Ferrand, France, 9-13 July 1990, 454-458. Available from WMO.
  • —Examination of how well Kalman filtering handles rapid developments of baroclinically unstable flows in the context of a simple model.
  • linden, P. 1990 (G). Methods of data assimilation. Meteorology and environmental sciences (R. Guzzi, A. Navarra and J. Shukla, eds.). World Scientific Publishing Co., Singapore, 121–165.
  • Warner, T. T. 1990 (G). Assimilation of data with mesoscale meteorological models. Proceedings, World Meteorological Organisation, International Symposium on Assimilation of observations in meteorology and oceanography, Clermont-Ferrand, France, 9-13 July 1990, 165-169. Available from WMO.
  • Bennett, A. F. and R. N. Miller, 1991 (A). Weighting initial conditions in variational assimilation schemes. Mon. Wea. Rev. 119, 1098–1102.
  • Blumenthal, M. B. 1991 (1). Predictability of a coupled ocean-atmosphere model. J. Climate 4, 766–784.
  • —Determination of optimal modes and their interpretation as a measure of predictability of ENSO.
  • Cohn, S. E. and D. F. Parrish, 1991 (K). The behaviour of forecast error covariances for a Kalman filter in two dimensions. Mon. Wea. Rev. 119, 1757–1785.
  • Daley, R. 1991 (KV). Atmospheric data analysis. Cambridge University Press, Cambridge, UK, 460 pp.
  • —Basic book. Discuss application of Kalman filter and variational methods to data assimilation.
  • Dee, D. P. 1991 (K). Simplification of the Kalman filter for meteorological data assimilation. Q. J. R. MeteoroL Soc. 117, 365–384.
  • Errico, R. M. and T. Vukicevic, 1991 (S). Sensitivity analysis using the adjoint of the PSU(NCA R mesoscale model Report No. 0501(91-3. NCAR, Boulder, CO. 38 pp.
  • —Determines the adjoint of a dry version of the model. Tests in cases of Alpine lee cyclogenesis, Atlantic explosive cyclogenesis. Compute the sensitivity of lowest surface pressure to initial state.
  • Fillion, L. 1991 (V). Variational implicit normal mode initialization on the sphere. Mon. Wea. Rev. 119, 631–652.
  • —Extension of the paper by Fillion and Temperton (1989).
  • Ghil, M. and P. Malanotte-Rizzoli, 1991 (KY). Data assimilation in meteorology and oceanography. Adv. Geophys. 33,141–266.
  • —General discussion of the relative merits of several approaches for data assimilation.
  • Hernandez, E., F. Martin and F. Valero, 1991 (S). State-space modelling for atmospheric pollution. J. App!. Meteor. 30, 793–811.
  • —Uses the adjoint technique with the model as a strong constraint.
  • Kapitza, H. 1991 (A). Numerical experiments with the adjoint of a nonhydrostatic mesoscale model. Mon. Wea. Rev. 119, 2993–3011.
  • —Assimilation of data from a single radar and discussion of the problem of ill-posedness in the absence of additional (temperature) information.
  • Louis, J.-F. 1991 (P). Use of the adjoint to optimize model parameters. Annales Geophysicae, Supplement to vol. 9, EGS 16th General Assembly, Wiesbaden, Germany, pp. C107.
  • Mahfouf, J.-F. 1991 (V). Analysis of soil moisture from near-surface parameters: a feasibility study. J. App!. Meteor. 30, 1534–1547.
  • — Comparison of variational and sequential algorithms. Obtains an estimate of soil-moisture from running a one-dimensional model and examining forecast errors of near-surface variables. Uses varia-tional and sequential (01) method. Tested on three 4-day forecasts for clear-sky conditions.
  • Rabier, F. and P. Bernardet, 1991 (V). Variational analysis of orographic waves. Beitr. Phys. Atmosph. 64, 207–217.
  • —Minimises the time tendency of the flow to retrieve steady orographic waves. Solves directly the Euler—Lagrange equations for a low-dimension model.
  • Smedstad, O. M. and J. J. O'Brien, 1991 (VP). Variational data assimilation and parameter estimation in the equatorial Pacific Ocean. Prog. Oceanog. 26, 179–241.
  • —Determines large scale spatial structure of the wave speed of a reduced gravity model, using data at 3 stations. Simulation experiments, and real data tests.
  • Sun, J., D. W. Flicker and D. K. Lilly, 1991 (V). Recovery of Three-Dimensional Wind and Temperature Fields from Single-Doppler Radar Data. Atmos. Sci. 48, 876–890.
  • —Boussinesq equations, dry model. Simulation test. Assumes reflectors that are passive tracers. Uses adjoint to compute the gradient of the cost function (radial wind error + reflectivity error). Studies the effect of amount, type and accuracy of observations. Results improved by adding a smoothness constraint. Uses an adjoint method.
  • Thepaut, J.-N. and P. Courtier, 1991 (V). Four-dimensional data assimilation using the adjoint of a mul-tilevel primitive equation model. Q. J. R. Meteorol. Soc. 117, 1225-1254 (also as Technical Memorandum No. 178. ECMWF, Reading, UK, 45 pp).
  • —Uses a 3D primitive equation model (the Météo-France (ECMWF ARPEGE/IFS model) without physics. Simulation experiments. Looks at controlling the gravity wave via Machenauer normal mode initialization, the impact of assimilation time interval, the impact of horizontal diffusion, and the distribution of observations in time.
  • Barkmeijer, J. 1992 (I). Local error growth in a barotropic model. Tellus 44A, 314–323.
  • — Optimal modes are examined for the inviscid barotropic vorticity equation and compared with the exponential modes, where growth is measured within various subdomains.
  • Bennett, A. F. and J. R. Baugh, 1992 (V). A parallel algorithm for variational assimilation in oceanography and meteorology. J. Atmospheric and Oceanic Technology 9, 426–433.
  • —Shows that the functional must include an explicitly contribution from the initial conditions.
  • Borges, M. D. and D. L. Hartmann, 1992 (I). Barotropic instability and optimal perturbations of observed nonzonal flows. J. Atmos. Sci. 49, 335–354.
  • — Determination and comparison of both optimal and exponential modes for a divergent barotropic model on the sphere.
  • Chao, W. C. and L.-P. Chang, 1992 (A). Development of a four-dimensional variational analysis system using the adjoint method at GLA. Part 1: Dynamics. Mon. Wea. Rev. 120, 1661–1673.
  • —Includes some interesting comments on adjoint testing.
  • Daley, R. 1992 (K). The effect of serially correlated observation and model error on atmospheric data assimilation. Mon. Wea. Rev. 120, 164–177.
  • —Examination of some assumptions commonly made in the application of Kalman filtering to atmo-spheric models.
  • Daley, R. 1992 (K). Forecast error statistics for homogeneous and inhomogeneous observation networks. Mon. Wea. Rev. 120, 627–643.
  • Daley, R. 1992 (K). Estimating model-error covariances for application to atmospheric data assimila-tion. Mon. Wea. Rev. 120, 1735–1746.
  • —Suggests a method for estimating the homogeneous, stationary component of the model-error covariances. Applied to a simple model.
  • Ehrendorfer, M. 1992 (K). Four-dimensional data assimilation: comparison of variational and sequential algorithms. Q. J. R. Meteorol. Soc. 118, 673–713.
  • —Kalman filtering and 4DVAR using adjoints are applied to the barotropic vorticity equation and compared with regard to accuracy and computational requirements. Discussion on benefits and drawbacks of the two methods.
  • Errico, R. M. and T. Vukicevic, 1992 (S). Sensitivity analysis using an adjoint of the PSU-NCAR mesoscale model. Mon. Wea. Rev. 120, 1644–1660.
  • —Computes the sensitivity of 36 hr forecasts at one point to the initial state.
  • Fillion, L. and M. Roch, 1992 (V). Variational implicit normal-mode initialization for a multi-level model. Mon. Wea. Rev. 120, 1050–1076.
  • — Extension of the paper by Fillion and Temperton (1989).
  • Gauthier, P. 1992 (A). Chaos and quadri-dimensional data assimilation: a study based on the Lorenz model. Tellus 44A, 2–17.
  • Hoffman, R. N., J.-F. Louis and T. Nehrkorn, 1992 (G). A method for implementing adjoint calculations in the discrete case. Technical Memorandum No. 184, ECMWF, Reading, Berks., UK, 19 pp.
  • Marais, C. and L. Musson-Genon, 1992 (P). Forecasting the surface weather elements with a local dynamical-adaptation method using a variational technique. Mon. Wea. Rev. 120, 1035–1049.
  • Marchuk, G. I. and Yu. N. Skiba, 1992 (S). Role of adjoint equations in estimating monthly mean air surface temperature anomalies. A tmósfera 5, 119–133.
  • Navon, I.M., X. Zhou, J. Derber and J. Sela, 1992 (A). Variational data assimilation with an adiabatic version of the NMC spectral model. Mon. Wea. Rev. 120, 1433–1446.
  • —Adiabatic NMC model at T40 truncation. Tests with simulated and real data. Effect of horizontal diffusion and surface drag.
  • Parrish, D. F. and J. C. Derber, 1992 (V). The National Meteorological Center's Spectral Statistical-Interpolation Analysis System. Mon. Wea. Rev. 120, 1747–1766.
  • — 3D variational analysis of meteorological fields.
  • Qiu, C.-J. and Q. Xu, 1992 (A). A simple adjoint method of wind analysis for single-Doppler data. J. Atmos. Oceanic Tech. 9, 588–598.
  • —Simulation study, using reflectivity advection equation as control equation. Retrieve the mean wind field over several scans.
  • Rabier, F. and P. Courtier, 1992 (A). Four-dimensional assimilation in the presence of baroclinic instability. Q. J. R. Meteorol. Soc. 118, 649–672.
  • —Shows the ability of four-dimensional variational assimilation to use the dynamical information for a typical baroclinic instability problem.
  • Rabier, F., P. Courtier and O. Talagrand, 1992 (S). On application of adjoint models to sensitivity analysis. Beitr. Phys. Atmosph. 65, No. 3, 177-192.
  • — Investigates the sensitivity of cyclogenesis with respect to initial conditions for a typical baroclinic instability problem. Applies non linear normal-mode initialization to eliminate the influence of gravity waves on the diagnostic function chosen to represent cyclogenesis.
  • Rabier, F. 1992 (ASPG). Assimilation variationnelle de donnees meteorologiques en presence d'instabilite barocline. PhD dissertation, University of Paris 6.
  • — Gathers all the author's publications on the subject.
  • Ramamurthy, M. K. and I. M. Navon, 1992 (A). The conjugate-gradient variational analysis and initialization method: an application to MONEX SOP 2 data. Mon. Wea. Rev. 120,2360–2377.
  • —A variational method for blending 2 analyses (produced at 2 different scales).
  • Robertson, A. W. 1992 (S). Diagnosis of regional monthly anomalies using the adjoint method. Part I: temperature. J. Atmos. Sci. 49, (11), 885–918.
  • —One-layer, tropospherically averaged tracer model for Central Europe. Looks at advection, adiabatic and diabatic heat sources.
  • Thépaut, J.-N. 1992 (AIG ). Application des methodes variationnelles pour l'assimilation quadridimension-nelle des observations météorologiques. PhD thesis, Université Paris 6,171 pp.
  • — Gathers all the author's publications on the subject.
  • Wang, Z., I. M. Navon, F. X. Ledimet and X. Zou, 1992 (G). The second-order adjoint analysis: theory and application. Meteorology and Atmospheric Physics 50,3–20.
  • Wergen, W. 1992 (P). The effect of model errors in variational assimilation. Tellus 44(A), 297–313.
  • —Uses a linearized, one-dimensional shallow water model. Introduces a phase speed error to explore the effect of model errors. Extends the assimilation method to retrieve a forcing term (simulating model physics). Compares to OI. Weak constraint optimization including the forcing terms gives the smallest analysis error, but the forecast degrades quickly because, in the assimilation, the forcing terms compensate for the phase speed error.
  • Zou, X., I. M. Navon and F. X. Le Dimet, 1992 (A). Incomplete observations and control of gravity waves in variational data assimilation. Tellus 44A, 272–296.
  • —The use of a penalty function applied to the geopotential tendency to reduce gravity wave oscillations in the results of 4DVAR is investigated. Also discussed are the impacts of incomplete observations on the solution convergence, uniqueness and quality.
  • Zou, X., I. M. Navon and F. X. Ledimet, 1992 (AP). An optimal nudging data assimilation scheme using parameter estimation. Q. J. Meteorol. Soc. 118,1163–1186.
  • Barkmeijer, J. 1993 (I). Local skill prediction for the ECMWF model using adjoint techniques. Mon. Wea. Rev. 121,1262–1268.
  • Gauthier, P., P. Courtier and P. Moll, 1993 (K). Assimilation of simulated wind lidar data with a Kalman filter. Mon. Wea. Rev. 121,1803–1820.
  • —Demonstrates how the variances of forecast errors can be modified by the dynamics in a barotropic model.
  • Molteni, F. and T. N. Palmer, 1993 (I). Predictability and finite-time instability of the northern winter circulation. Q. J. R. Meteorol. Soc. 119,269–298.
  • —Computes most unstable modes of a baroclinic quasi-geostrophic model.
  • Rabier, F., P. Courtier, J. Pailleux, 0. Talagrand and D. Vasiljevic, 1993 (A). A comparison between four-dimensional variational assimilation and simplified sequential assimilation relying on three-dimensional analysis. Q. J. R. Meteorol. Soc., July issue.
  • Rinne, J. and H. Järvinnen, 1993 (P). Estimation of the Cressman term for a barotropic model through optimization with the use of the adjoint model. Mon. Wea. Rev. 121,825–833.
  • Thépaut, J.-N., D. Vasiljevic, P. Courtier and J. Pailleux, 1993 (A). Variational assimilation of conven-tional meteorological observations with a multilevel primitive equation model. Q. J. R. Meteorol. Soc. 119,153–186.
  • —Use of real data, study of interactions between observations and dynamics. The implicit use of flow dependent structure functions is studied. 4D-Var is compared to Optimal Interpolation.
  • Zou, X., I. M. Navon, F.-X. Le Dimet, M. Berger, M. K. Phua and T. Schlick, 1993 (G). Numerical experience with limited memory quasi-Newton methods for large-scale unconstrained nonlinear minimization. SIAM J. Optimiz., in press.
  • — Review of adjoint method for data assimilation. Tests of different minimization algorithms. Uses shallow water equations.
  • Zou, X., I. M. Navon and J. Sela, 1993 (A). Control of Gravitational Oscillations in variational data assimilation. Mon. Wea. Rev. 121, 272–289.
  • —Confirms with a more detailed study the results on the control of the gravity waves in a 3D model obtained by Thépaut and Courtier (1991).

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