Uniied Notation for Data Assimilation: Operational, Sequential and Variational 1

The need for uni ed notation in atmospheric and oceanic data assimilation arises from the eld s rapid theoretical expansion and the desire to translate it into practical applications Self consistent notation is proposed that bridges sequential and variational methods on the one hand and operational usage on the other Over various other mottoes for this risky endeavor the authors selected When I use a word Humpty Dumpty said in rather a scornful voice tone it means just what I choose it to mean neither more nor less Lewis Carroll J Met Soc Japan Special Issue on Data Assimilation in Meteorology and Oceanography Theory and Practice Vol No B pp Corresponding author Current a liation CNES place Maurice Quentin Paris Cedex France J Met Soc Japan K Ide P Courtier M Ghil and A C Lorenc Introduction and motivation Model based assimilation of observations or data assimilation for short has evolved into a major area of academic research in dynamic meteorology and physical oceanography as well as of operational numerical prediction for atmospheric and oceanic ows Panel Its major sources of theoretical and practical ideas include by now the engineering Bucy and Joseph Gelb Jazwinski mathematical Gill et al Lions Marchuk and geophysical Bennett Daley Ghil and Malanotte Rizzoli literatures The Second WMO International Symposium on Assimilation of Observa tions in Meteorology and Oceanography held in Tokyo in March has shown substantial progress in the practical implementation of advanced data assimilation methods based on both the control theoretical and sequential estimation approach These two approaches be cause of their di erent heritage have sometimes used very di erent sets of notation which obscure the fact that they attempt to solve in complementary ways similar problems This has hindered mutual understanding amongst developers and users of data assimilation meth ods provided a stumbling block in the operational application of novel methods based on either approach or a combination of both and hampered further theoretical advances Notation is the language of the sciences and di ering sources of notation guarantee a Tower of Babel that impedes e ective communication amongst the theoreticians and prac titioners of data assimilation It also renders the literature more di cult to master for newcomers whether students or researchers from other elds The authors are keenly aware that no single notation is ideal and can make every user happy This modest proposal is made with the hope that it will catch on and thereby help advance data assimilation in both meteorology and oceanography by facilitating the mastery of past and present methods based on either approach and fostering the development of future more advanced methods based on either combinations of these two approaches or on entirely novel approaches Basic concepts and de nitions appear in Section Recommended self consistent nota tion following the sequential approach is outlined in Section and following the variational approach in Section Methods based on various combinations of these two approaches are under investigation and we hope to contribute to the study and implementation of such com bined methods by suggesting uni ed notation Conclusions and a summary table follow in J Met Soc Japan K Ide P Courtier M Ghil and A C Lorenc Section An Appendix includes tentative recommendations for continuous space and time notation at one of the research frontiers of the eld This paper is restricted by its nature and limitations of space to present basic concepts and notation which are merely illustrated by a few examples For a more exhaustive presentation of the rapidly increasing number of assimilation methods and their complete description we refer to the books and review papers already cited and to those included in this volume or cited below Basic concepts and de nitions A discrete model for the evolution of an atmospheric oceanic or coupled system from time ti to time ti is governed by an equation x ti Mi x f ti where x and M are the model s state vector and its corresponding dynamics operator re spectively The state vector x has dimension n The dynamics M of the model evolution in a computer simulation or prediction is commonly nonlinear and deterministic see also the Appendix while the true geo uid may di er from by random or systematic errors The state vector x is obtained usually by discretization of the full partial di er ential equations that we assume govern the ow using nite di erences nite elements or pseudo spectral methods The error covariance matrix associated with x is given by P The theoretically interesting case of continuous rather than discrete space and time is treated in the Appendix Observations y at time ti are de ned by y i Hi x t ti i where H is an observation operator and is a noise process The observation vector y i y ti has dimension pi A major problem of data assimilation is that typically pi n The operator H can also be nonlinear like M and both can contain explicit time dependence denoted here by the subscript i in addition to the implicit dependence via the state vector xi x f ti The noise process is commonly assumed to have zero mean and we denote its covariance matrix by R it consists of instrumental and representativeness errors whose covariance matrices are E and F respectively with R E F More generally i J Met Soc Japan K Ide P Courtier M Ghil and A C Lorenc may have a nonzero time and state dependent mean and the covariance matrices E F and R may all depend on ti or xi In general subscripts are used to i denote the discrete time index as in Eqs above ii refer to the corresponding space related indices on a regular or irregular grid or for spectral coe cients and iii indicate in parentheses the vector e g state or observation with respect to which an error covariance matrix is de ned see Section for an example Superscripts on the other hand are used to i describe mathematical operations such as T I and for transpose inverse generalized inverse and linearization re spectively and ii refer to the nature of vectors or matrices in the data assimilation process such as a b f and o for analysis background often called rst guess but see Remarks and below forecast and observation Upper and lower case char acters in superscripts indicate i some mathematical operations and ii the type of the variable respectively Furthermore may be dropped when the original operator is linear