Documentation of the Physical Space Statistical Analysis System PSAS Part II The Factored Operator Formulation of Error Covariances

This article is a theoretical basis for the software implementation of the Physical space Statistical Analysis System PSAS that is used for atmospheric data analysis at the NASA Data Assimilation O ce DAO The PSAS implements a statistical algo rithm that combines irregularly spaced observations with a gridded forecast to produce an optimal estimate of the state of the atmosphere Starting frommodels for the forecast and observation errors the PSAS version v uses a factored operator formulation for the error covariance matrices This formulation determines how the observational data and their attributes as well as the error covariance matrices are managed during the life cycle of the algorithm this is the main source of software complexity of the PSAS This is mainly due to the diversity of data types and sources as well as the use of the multivariate formulation as described in the text The coordinate systems and data types used in the PSAS analysis are described The PSAS univariate forecast error covariance models are introduced and the multivariate upper air and sea level coupled height wind and decoupled wind forecast error covariance models are derived from the univariate height forecast error covariance models The factorization of these multi variate covariance models into a product of matrices is described Observation error covariances used in the PSAS are brie y discussed Finally we discuss the structure of the matrices in the software implementation of the PSAS and some related software issues An on line version of this document can be obtained from ftp dao gsfc nasa gov pub office notes on v ps Z postscript Visit also the Data Assimilation O ce s Home Page at http dao gsfc nasa gov