The paper proposes a unified notation for data assimilation in atmospheric and oceanic sciences to bridge sequential and variational methods, as well as operational usage. It addresses the need for a consistent notation due to the rapid theoretical development and practical applications of data assimilation. The authors emphasize the importance of a common language to facilitate communication among researchers and practitioners, avoiding the confusion caused by differing notations that hinder understanding and implementation of methods. The paper outlines basic concepts and definitions, including the model dynamics, observation operators, and error covariance matrices. It then presents recommended, self-consistent notations for sequential and variational methods. Sequential methods, such as the Extended Kalman Filter (EKF) and Optimal Interpolation (OI), are discussed, along with their respective notations. Variational methods, including 4D-Var and incremental 4D-Var, are also introduced, with their notations and applications explained. The paper highlights the differences and similarities between sequential and variational approaches, emphasizing the need for a unified notation to support both theoretical and practical developments. It also discusses the challenges and considerations in using these methods, including computational efficiency, error covariance matrices, and the interpretation of different variables. The authors conclude that a consistent set of symbols is essential for advanced assimilation methods, and that unified notation facilitates the use of these methods in both sequential and variational approaches. The paper provides a table of notations by type, including vectors, operators, matrices, superscripts, and subscripts, to guide the use of notation in data assimilation. The paper also references earlier works that have contributed to the field of data assimilation, emphasizing the importance of a standardized notation for effective communication and application.The paper proposes a unified notation for data assimilation in atmospheric and oceanic sciences to bridge sequential and variational methods, as well as operational usage. It addresses the need for a consistent notation due to the rapid theoretical development and practical applications of data assimilation. The authors emphasize the importance of a common language to facilitate communication among researchers and practitioners, avoiding the confusion caused by differing notations that hinder understanding and implementation of methods. The paper outlines basic concepts and definitions, including the model dynamics, observation operators, and error covariance matrices. It then presents recommended, self-consistent notations for sequential and variational methods. Sequential methods, such as the Extended Kalman Filter (EKF) and Optimal Interpolation (OI), are discussed, along with their respective notations. Variational methods, including 4D-Var and incremental 4D-Var, are also introduced, with their notations and applications explained. The paper highlights the differences and similarities between sequential and variational approaches, emphasizing the need for a unified notation to support both theoretical and practical developments. It also discusses the challenges and considerations in using these methods, including computational efficiency, error covariance matrices, and the interpretation of different variables. The authors conclude that a consistent set of symbols is essential for advanced assimilation methods, and that unified notation facilitates the use of these methods in both sequential and variational approaches. The paper provides a table of notations by type, including vectors, operators, matrices, superscripts, and subscripts, to guide the use of notation in data assimilation. The paper also references earlier works that have contributed to the field of data assimilation, emphasizing the importance of a standardized notation for effective communication and application.