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We present an innovative approach for solving time dependent Four Dimensional Variational Data Assimilation (4D VAR DA) problems. The proposed approach performs a decomposition of the whole physical domain, i.e. both along spatial and temporal directions; a reduction in space and time of both the Partial Differential Equations-based model and the Data Assimilation functional; finally it uses a modified regularization functional describing restricted 4D VAR DA problems on the domain decomposition. Innovation mainly lies in the introduction ab initio, i.e. on the numerical model – of a domain decomposition approach in space and time joining the idea of Schwarz's method and Parallel in Time (PinT)–based approaches. We provide the numerical framework of this method including convergence analysis and error propagation. A validation analysis is performed discussing computational results on a case study relying on Shallow Water Equations. [ABSTRACT FROM AUTHOR]